A HISTORY OF FINITE SIMPLE GROUPS
by
Faun C.C. Doherty
B.A., Oberlin College, OH, 1993
; A thesis submitted to the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Master of Science
 Applied Mathematics
1997
This thesis for the Master of Science
degree by
Faun C.C. Doherty
has been approved
by
Date_____
Doherty, Faun C.C. (M.S., Applied Mathematics)
A History of Finite Simple Groups
Thesis directed by Associate Professor J. Richard Lundgren
ABSTRACT
A group is a set together with an associative binary operation such
that there exists an identity element for the set, and an inverse for each element
in the set. All finite groups can be broken down into a series of finite simple
groups which have been called the building blocks of finite groups. The
history of finite simple groups originates in the 1830s with Evariste Galois and
the solution of fifth degree polynomial equations. In the twentieth century, the
recognition of the importance of finite simple groups inspired a huge effort to
find all finite simple groups. This classification project was completed in 1981.
We shall begin by taking a historical look at the earliest methods of analyzing
the structure of finite groups according to their order. Finite simple groups
can be divided into two types, those belonging to infinite families and the 26
sporadic simple groups. We shall look at the discovery and representation of
many of these. Finally, we shall discuss the monumental 10,000 to 15,000 page
proof of the classification of all finite simple groups.
This abstract accurately represents the content of the candidates
thesis. I recommend its publication.
Signed
J. Richard Lundgren
in
ACKNOWLEDGEMENTS
I would like to thank Professor Lundgren for his support in writing
this thesis. Also, thanks to my parents for their example and Michael for his
patience.
CONTENTS
Chapter
1 Introduction.................................................... 1
2 The Range Problem............................................... 3
2.1 Introduction to the Problem ............................. 3
2.1.1 Sylows Theorems......................................... 4
2.1.2 Other Theorems, Corollaries, Etc. That Will Prove
Useful:.................................................. 5
2.2 Some History............................................. 7
2.2.1 Holder................................................... 7
2.2.2 Cole, Burnside.......................................... 12
2.2.3 The Completion of the Range Problem Through Order
One Million ............................................ 16
2.3 Some Examples .......................................... 19
3 The Simple Groups ............................................ 30
3.1 Infinite Families of Simple Groups........................ 30
3.1.1 The Alternating Groups.................................. 30
3.1.2 Simple Groups of Lie Type............................... 35
3.1.2.1 The Classical Linear Groups......................... 36
3.1.2.2 Other Lie Groups................................... 39
3.2 The Sporadic Simple Groups............................. 40
v
3.2.1 The Mathieu Groups.................................... 42
3.2.2 Centralizer of Involution Problems.................... 43
3.2.3 Rank 3 Permutation Groups............................. 48
3.2.4 The Remaining Sporadic Simple Groups.................. 49
4 The Classification Theorem................................... 51
4.1 History................................................. 51
4.2 The Theorem............................................. 56
References........................................................... 63
vi
1. Introduction
Some have referred to the study of simple groups as the El Dorado
of finite group theory. It has been a very active field of study through the
twentieth century and has its roots in the nineteenth, as does group theory
itself. A group is defined as a set together with an associative binary operation
defined such that there exist an identity element for the set, and inverses for
each element of the set. The set is closed under the operation. A normal
subgroup H of a group G is a subgroup such that aH = Ha for all a E G.
Another definition of normal is that a1 Ha = H for a G G. A simple group is
a group which has no normal subgroups except itself and the identity (which
are always normal). Those groups with prime order have no subgroups except
for the identity and the group itself, thus they are considered trivially simple.
For the rest of this paper, the term simple group will refer to finite nontrivial
simple groups. Simple groups are special kinds of groups that are the building
blocks of all other groups, thus the importance in their study. This idea was
recognized as early as 1832 by Evariste Galois, and later a search for the
simple groups took place. In the twentieth century this search culminated in
a monumental theorem which classifies all simple groups. One of the earliest
methods of locating simple groups is called the range problem. This is a
systematic examination of the internal structure of groups according to the
order of the group. Chapter one of this paper will outline the history of this
1
problem and the methods used through the analysis of groups up to order one
million. The second chapter will describe the simple groups by their types:
infinite families of simple groups and the sporadic simple groups. How some
of these groups can be represented as well as the methods of their discovery
will be discussed. Finally, a general outline of the classification theorem will
be given in the last chapter.
2
2. The Range Problem
2.1 Introduction to the Problem
Among the methods of determining all finite simple groups, the ap
proach of examining individual groups of certain orders can seem at times slow
and methodical. Yet this task, begun in 1892 by Otto Holder, has proven fruit
ful in the advancement of group theory, if not always in the discovery of new
simple groups. It has shed a great deal of light upon the structure of groups
with given orders which allows one to understand the nature of simple groups,
at least in so far as determining what they are not. This particular prob
lem lasted through to 1975 when Marshall Hall, Jr. completed the individual
examination of groups with particular orders through the order of 1,000,000.
About eleven individuals from 1892 to 1975 participated in the solution of this
problem, each aided by the work and discoveries of those who came before.
The range problem itself is not difficult to understand, in light of the
search for simple groups. It is simply this: given a particular natural number,
say n, what can we say about the structure of any group having n elements?
And in particular, can we determine if the group has any normal subgroups
besides itself and the identity, i.e., can we show that the group is not simple?
If the group is simple, is it unique? Through the history of this problem,
there were two main methods used to explore the structure of groups with a
given order. One was to use the Sylow theorems and the other was to employ
3
character theory. It will be the task of this paper to concentrate only on the
Sylow theorem methods, thus a word about these theorems is in order.
2.1.1 Sylows Theorems
Ludvig Sylow, a Norwegian mathematician came up with the Sylow
theorems in 1872 by way of the study of permutation group theory. These
results lost no importance with the development of abstract group theory, in
fact, their importance grew. The Sylow theorems as we state and prove them
today are based on the fundamental concept known as Lagranges theorem,
and it is here that we shall start.
Theorem 2.1 [Lagranges theorem] Suppose H C G is a subgroup. Then
\G\ = \H\\G:H\.
Note that G : H\ is the index of H in G, or the number of distinct
right cosets of H in G. A right coset is the set Hg = {hg \ h H} where
H C G. We can show easily that the group G is the disjoint union of the
distinct right cosets. The cardinality of each coset is equal to the number of
elements in the subgroup H, and with these two facts, we may deduce that
the order of G is the order of H times the number of distinct right cosets that
partition G.
Theorem 2.2 [Sylows Theorem 1] If pk \ \G\, then G has at least one
subgroup of order pk for any prime p.
Thus if any power of a prime divides the order of our group, then the
group has a subgroup of order that power of the prime.
Theorem 2.3 [Sylows Theorem 2] If H C G, and \H\ = pk then H is
4
contained, in some Sylow psubgroup.
A Sylow psubgroup is a subgroup of G such that its order is equal
to the full power of p in the order of G. For example, if we have a group of
order 24 5 ll2, a sylow 2subgroup would have order 24. The set of all Sylow
psubgroups of G is denoted Sylp(G). We know from the first Sylow theorem
that Sylp(G) is not empty. We can also find a Corollary ((2.8) below) which
states that if only one Sylow psubgroup exists, then it is normal in G. This
fact will allow us to eliminate easily many integers as possible orders of simple
groups.
Theorem 2.4 [Sylows Theorem 3] The number of Sylow psubgroups of G,
i.e., Sylp(G)\(written np) has the following properties:
np = 1 mod p
and np = 1 mod pe
if pe < \S : S fl T\ for all S and T e Sylp(G) with S ^ T.
The examination of the structure of groups with a given order is feasible be
cause of a number of other results besides the Sylow theorems, although many
of these results are based on the Sylow theorems. A number of these results
shall be listed below and referred to throughout this chapter.
2.1.2 Other Theorems, Corollaries, Etc. That Will Prove Useful:
Theorem 2.5 A nontrivial finite pgroup has a nontrivial center.
A pgroup (where p is prime) is defined as a group in which every element has
order a power of p. The center of a group (Z(G)) is a normal subgroup of G
composed of all elements which commute with all other elements of G.
5
Theorem 2.6 If G = pa, where p is prime and a > 1, then G is not simple.
Proof: Let G = pa, and suppose that G is simple. Since G itself is apgroup,
by (2.5) we know that 1 < Z(G) < G, and since G is simple, Z(G) must be
G. But then G is abelian, and its simplicity implies that G = p. This is a
contradiction, since G = pa, so G is not simple.
Theorem 2.7 [The N/C Theorem] If H C G, then the factor group of the
normalizer of H in G by the centralizer of H in G is isomorphic to a sub
group of the group of all automorphisms of H. In mathematical notation,
Ng{H)/Cg{H) M where M C Aut(H).
Corollary 2.8 A unique Sylow psubgroup is normal.
Lemma 2.9 Let G = pam, where a > 0, m > 1 and p does not divide m. If
G is simple, then np{G) satisfies all of the following:
1. np divides m
2. np = 1 mod p
3. G divides (np\)
Corollary 2.10 Let P be a Sylow psubgroup of G. Then np = \G : NG(P) \,
and np divides \G \ P\.
Theorem 2.11 Let H C G with \G \ H\= n. Then there exists N < G such
that N C H and \G : N\ divides n\. In particular, if n> 1 and G does not
divide n\, G is not simple.
Corollary 2.12 Let H C G and \G : H\ = p where p is the smallest prime
divisor of G. Then H
Theorem 2.13 Let B and C be cyclic of order n < oo. Then B = C and
6
there are exactly (p(n) different isomorphisms that map B to C.
Theorem 2.14 Every two Sylow psubgroups of G are conjugate.
2.2 Some History
2.2.1 Holder
The range problem was initiated by Otto Holder (18591937) in 1892.
Before 1892, Holder published two papers that considerably contributed to the
emphasis on this problem. The first was published in 1889 in the Mathema
tische Annalen [15]. It was a paper primarily dealing with the solution of
equations. However, what was evolving into group theory, thanks to Evariste
Galois, who we will discuss in chapter 2, seems to have proved useful to his
work. The concepts of normal subgroups and a composition series are dis
cussed. A composition series is a series of normal subgroups
1 = Gq <] Gi <]...<] Gni o Gn = G
where no normal subgroups exist between each Gi (i.e., each subgroup is max
imal normal in the next.) The factor groups Gi/Gii are all simple groups.
What is so important about this series is expressed in the JordanHolder the
orem which states that these simple groups (called composition factors) are
uniquely determined up to isomorphism. Thus, it is apparent that the compo
sition series acts as a type of fingerprint for a group. Holder was among the
first to recognize that the composition factors are building blocks of groups,
and deserve special study. It was the second paper, published in 1892, in which
Holder states
It would be of the greatest interest if a survey of all simple groups with
7
a finite number of operations [elements] could be known [15].
One other advancement of this time deserves recognition. Group
theory was evolving into a subject in its own right, and the idea of treating
groups in the abstract, an idea attributed to Cayley, was finally being accepted.
In his start upon the range problem, Holder was the first to study groups in
the abstract. More often in the past, groups were considered with respect to
their mode of representation, for example, a linear transformation group. The
range problem initiated the type of exploration that only required knowledge
of the order of the group.
Holder studied groups having orders from 1 to 200. He did not dis
cover any new simple groups, since the unique simple group of order 60 was
known to be simple (it is A5 and will be discussed below), as was the group of
order 168 (PSL2(7), found by Jordan in 1870). His methods were important,
however, since they were used by all others working later on the range prob
lem. His ideas provided important general theorems which can be, and were
used within any range, and will be discussed below.
The most useful tools that Holder employed were the Sylow theorems.
Holder was comfortable with permutation groups, and also used this theory.
Many of the lemmas that he used in more general theorems came from permu
tation group theory combined with the results of Sylows theorems. One of his
general theorems has to do with groups that have orders equal to a product of
three or fewer primes, not necessarily distinct. Holder proved that groups with
orders pq, p2q, or pqr are not simple. Sylow had already taken care of those
groups with orders pa (2.6). These theorems can be proven in a more effective
8
manner using only the Sylow theorems, which Burnside did in later years. The
following proofs are similar to the methods used by Burnside, rather than the
permutation theory used by Holder.
Holders Proofs Using the Sylow Theorems
Theorem 2.15 If \G\ = pq, where p and q are primes, then G is not simple.
Proof: Let G = pq, where p and q are primes, and assume G is simple.
Without loss of generality, we may assume that p > q. Then the only choice
for np is np q, since np must divide q by (2.9), but cannot equal 1 by (2.8)
and our assumption. This implies that q = 1 mod p, which is a contradiction
since p> q. Thus, our assumption is false, and G is not simple.
Theorem 2.16 If G = p2q, where p and q are primes, then G is not simple.
Proof: Let G = p2q, where p and q are primes, and assume G is simple. The
choices for nq are: nq = p or p2. Suppose that nq = p. Then p = 1 mod q so
p> q. But the only choice for np is q which implies q = 1 mod p, thus, q > p.
So nq^p, which means that nq=p2. Let us now count elements in the group.
Since nq = p2, we have p2 subgroups each with order q. Notice that they have
prime order, which means that they are cyclic, and have no two elements in
common except for the identity. This means that there are p2(q 1) elements
with order q. Let denote the number of the rest of the elements. Then
\G\ = e+p2{q 1), or =p2qp2{q~ 1) =p2.
Thus, there are enough elements not of order q to only fit into one Sylow
psubgroup, which means there is a unique Sylow psubgroup, which must be
normal in G. But this is a contradiction to our assumption that G is simple,
which leaves us only with the alternative that one of the Sylow subgroups is
9
unique, thus normal (2.8). Thus, our assumption was wrong, and G is not
simple.
Theorem 2.17 If G '= pqr, where p q and r are primes, then G is not
simple.
Proof: Assume G = pqr, where p, q and r are primes, and assume G is
simple. Without loss of generality, we may assume that p > q > r. The
possibilities for the size of Sylp(G) are as follows:
np = q, r, or qr
nq = p, r, or pr
nr = p, q, or pq.
Notice that we may eliminate q and r as possibilities for np since p > q > r
(using Sylows 3rd (2.4)). Also, we may eliminate r as possibility for nq for
the same reasons. We may eliminate q as possibility for nr since we know that
G cannot divide q\ since there is no p factor in q\. We conclude that there
are four cases only:
np = qr np = qr np = qr np = qr
1 nq=p 2. nq=p 3. nq = pr 4. nq = pr
nr = p nr = pq nr = p nr = pq.
If we examine each of these cases by counting elements, we find that none are
feasible.
The First Case: we can conclude that the number of elements with
order p is qr(p 1), the number of elements with order q is p(q 1) and
the number of elements with order r is p(r 1). Note that this is possible
since each Sylow subgroup has prime order, so no two Sylow subgroups of the
10
same order have elements in common except for the identity. If we add the
number of elements that we have so far, it is qr(p 1) + p(q 1)+ p(r 1) =
pqr qr + pq p + pr p or, pqr qr +pq + pr 2p Note that qr + pq is
positive since p > r, and pr 2p is positive if r > 2 (zero otherwise). Thus,
we have pqr+ some positive number as the number of elements in G, which is
a contradiction. Thus, the number of Sylow subgroups is not the first case.
The Second Case: Using the same arguments as above, the second
case provides us with the following number of elements: qr(p 1) +p(q 1) +
pq(r 1) but this is equal to pqr qr+pqp+pqr pq = pqr + pqr qr p
or, pqr + qr{p 1) p and we see that qr(p 1) p must be positive. Thus,
again we have over pqr number of elements, which is a contradiction.
The Third Case.'Using the same arguments as above, the third case
provides us the following number of elements: qr(p 1) +pr(q 1) +p(r 1)
which is equal to pqr qr + pqr pr +pr p = pqr + pqr qr p and this
is identical with the second case, and thus a contradiction.
The Fourth Case: The same counting technique provides us with the
following number of elements: qr(p 1) +pr(q 1) +pq(r 1) which equals
pqr qr + pqr pr + pqr pq = pqr + pqr qr + pqr p(q + r) and note
that'pgr qr is positive, as is pqr p(q + r) if q + r < qr, which is true if
q and r are > 3 and 2 respectively, which they are. Thus we have another
contradiction, which implies that our original hypothesis was incorrect, and G
is not a simple group.
The power of these theorems, along with a few others, eliminated all
but seven orders out of the first 200 cases. The seven remaining groups had
11
orders 60 (known to be simple), 90, 112, 120, 144, 168 (known to be simple),
and 180. Holder was able to show that all but 60 and 168 were orders of non
simple groups using various techniques of permutation group theory, yet it has
been said that his ability to use permutation groups was somewhat lacking. It
did take him nearly twenty pages of calculation to demonstrate that groups of
order 144 and 180 were not simple.
2.2.2 Cole, Burnside
It was an American mathematician who followed the path laid by
Holder. Frank Nelson Cole (18611927) continued the range problem in 1892
93 examining groups with orders ranging from 201 to 660. The methods used
by Holder were also used by Cole. The Sylow theorems provided the most
powerful tool of investigation, and Cole also looked at groups in the abstract
sense, only recurring when convenient to their representation in terms of sub
stitutions of n letters [permutation groups] [15]. Holders theorems of three
or fewer primes proved useful to eliminate all but 84 groups between 201 and
500. Sylows theorem that np = 1 mod p eliminated another 56. Eventually,
Cole determined that A6, PSL2( 11), and PSL2(23), groups of orders 360, 660
and 504 respectively, were the only simple groups with order between 201 and
660. The simple group of order 504 was never recognized as simple before
Coles work, even though it had been discussed by mathematicians such as
Mathieu and Kirkman. It was classified later as PSL2(23) following the ad
vancements made by Dickson and Moore. It was a special discovery in more
12
ways than one, since it launched the work of Eliakim Hastings Moore (1862
1932) who discovered that the infinite family of groups, PSL2(pn) was simple
except when pn 2 or 3. This in turn led to the proof by Dickson that the
infinite family of groups PSLm(pn) are simple, which is a generalization of
Jordans original 1870 result. This family shall be discussed further in the
subsequent chapter. Notice that there were no new methods evident in Coles
work, the Sylow theorems served him well.
William Burnside (18521927), who has been called the first real
group theorist in history because of his dedication to abstraction, was the next
mathematician to work on the range problem. Once again, his techniques did
not stray far from the Sylow theorems and permutation group theory. He
did develop some arithmetic tests, the most important of which states that
a simple group of even order must be divisible by either 12, 16, or 56. The
understanding of permutation groups had advanced since Holders and Coles
work, which was a help in Burnsides pursuits. Ironically though, Burnside
was very active in rewriting theorems previously based on permutation theory
using only the abstract ideas such as conjugacy classes and normalizers. Burn
side claimed that even in reference to the proofs of the Sylow theorems, from
the point of view of the right method they leave something to be desired [15].
He subsequently rewrote them. Notice that the proofs given above of Holders
three or fewer primes theorem are essentially Burnsides rewrites. Not only
did Burnside simplify the proofs for these, but he also extended the theorem
to include combinations of four or fewer primes. A couple of his proofs are
given below.
13
Burnside Theorems of Four or Fewer Primes
Theorem 2.18 If (7 = p3q where p and q are primes, then G is not simple.
Proof: Assume G = p3q, where p and q are primes, and assume G is simple.
The choices for nq are: nq= p p2, or p3, and nv = q, which implies
that q > p. Suppose that nq p. Then p = 1 mod q which contradicts q > p.
Suppose that nq = p3. Count elements: there are p3 subgroups, each with
order q, which have trivial intersections. Thus there are p3(q 1) elements
with order q. Let 0 denote the number of the rest of the elements. Then
G = 0 + p3(q ~ 1), or 0 = p3q p3(q 1) = p3.
Thus, there are enough elements not of order q to only fit into one
Sylow psubgroup, which means there is a unique Sylow psubgroup, which
must be normal in G. But this is a contradiction to our assumption that G is
simple, which leaves us with the last possibility.
Suppose that nq = p2. Then p2 = 1 mod q => q \ (p2 1) => q \
(p +1) (p 1). Since q is prime, this implies that q\ (p +1) or q \ (p 1). Since
q > p q  (p + 1) only. But this implies that p < q < p + 1 so q = p+l and
p and q are consecutive primes. But the only consecutive primes are 2 and 3,
so if G is indeed simple, p = 2 and q = 3 is the only possibility. Thus, if we
show that a group of order 23 3 is not simple, we have a contradiction.
Suppose G = 23 3. Then ri2 = 1 or 3. But note that G does
not divide 3!. Thus 712 = 1, which implies that G is not simple. So the only
possibilities left are that one of the original Sylow subgroups is unique, thus
normal, contradicting our hypothesis that G is simple.
Theorem 2.19 If G = p2q2 where p and q are primes, then G is not
14
simple.
Proof: Assume G = p2q2 where p and q are primes, and assume G is simple.
Without loss of generality, we may assume that q > p. The choices
for nq are: nq=p and p2. If nq = p, then p = 1 mod q, which contradicts the
fact that q > p.
So suppose that nq p2. Then the argument from the above proof
holds, i.e., p2 = 1 mod q => q  (p2 1) => q \ (p + l)(p 1). Since q is prime,
this implies that q  (p + 1) or q \ (p 1). Since q > p q \ (p+ 1) only. But
this implies that p
order 22 32 is not simple:
Let G = 22 32 Then n3 = 1,2, or 4. But 2 is not = 1 mod 3, and
<31 does not divide 4!, thus the Sylow 3subgroup is unique, thus normal in
G. So the only possibilities left are that one of the original Sylow subgroups is
unique, thus normal, contradicting our hypothesis that G is simple.
In 1895, Burnside completed the range problem up to order 1092.
Shortly after this time, beginning in 1896, a new technique emerged developed
by Burnside and Georg Frobenius (18491917) called character theory. This
theory, which is based on the study of certain functions (characters) from
a group into the complex numbers, has made a great impact on the study
of simple groups through this century. It was character theory that provided
Burnside with a proof of a monumental theorem that follows and outshines
the four or fewer primes result. In 1904, Burnside proved that any group with
order paqb where p and q are prime is not simple, unless it is of prime order
[6]. Obviously, this theorem plays a significant role in the simplification of the
15
work required on the range problem after 1904. The extension of this result to
orders made up of a combination of three primes, paqbrc, has been a difficult
problem which has lasted until the present, and the method of investigation
has most often been character theory. Unfortunately, it is beyond the scope
of this paper to mention character theory in more depth.
2.2.3 The Completion of the Range Problem Through Order One
Million
The turn of the century saw two mathematicians, George Abram
Miller (18631951) and his student G. H. Ling work on the range problem for
orders between 1092 and 2001 in 1900. The original techniques of investigation
had not changed much, however there were a couple of new results which came
from the older methods. One was that any group of order paq, paq2, and paqb
(for a = 1,...,5,p < q) was not simple. Notice that these results were the
previews of what was to come in Burnsides 1904 theorem. The problem of
odd versus even orders was well under investigation at this time, as we shall
examine in the next chapter. The result at this time which was put to good
use was the fact that there were no simple groups with odd orders less than
2835. There was increased work on the theory of permutation groups, and on
transitive groups in particular which helped with the investigation of individual
orders. A permutation group on a set is called transitive if for each pair of
elements of the set, there exists an element in G which sends one to the other.
With these techniques, Miller and Ling showed that there was no simple group
between 1093 and 2000. There seemed to be quite a gap after the work of Miller
16
in interest in the range problem. It was not until 1912 that anyone approached
the orders following 2000. This may have been due to the difficulty that the
larger orders presented, and the lack of new results which would act quickly
and sweepingly, although one must remember the Burnside theorem which did
exactly that. It was not until 1954 that new methods actually arose to handle
groups of particular orders.
While work on the infinite families of simple groups was taking place,
there was a bit of a lull in the advances on the range problem during the early
twentieth century. In fact, work on the range problem was sporadic through
the twentieth century. L. P. Siceloff was the next mathematician to tackle the
orders 2001 through 3640 in 1912. He found simple groups with orders 2448,
2520 and 3420. He was not able to prove the uniqueness of the simple group
with order 2520, and it was not until 1922 that Miller successfully showed
that the group was Aj and unique. Cole came back to the game in 1924 with
the orders 3641 through 6232. He found four simple groups having orders
3420, 4080, 5616, and 6048. He found difficulty with the uniqueness of two
orders, 5616 and 6048. Both of these are unique simple groups, as shown by
Richard Brauer in 1942 using character theory. It took eighteen years to find
the methods to complete this task! The next time that someone chipped away
at the range problem was in 1963. Michaels took the task of showing that
the unique simple groups between 6233 and 20,000 were of orders 7800, 7920,
9828, 12,180, and 14,880.
In 1972, Marshall Hall, Jr (19101990) extended the range problem
to order one million [12]. He drew together all of the methods used from the
17
late nineteenth century onwards, a great deal of the later methods relying
on advanced techniques of character theory. His assortment of methods also
included some computer work. Halls methods were unsuccessful with only
21 orders. It was in 1975 that two students, Beisiegel and Stingl, extended
work on the classification of simple groups according to the size of their Sylow
2subgroups undertaken by Paul Fong. The remaining 21 orders were taken
care of, and the range problem to one million was complete.
It was not necessarily the people working on the range problem that
discovered new simple groups. In fact, not many new simple groups were
found at all during the course of the range problem. In 1900, Dickson listed a
total of 53 known simple groups, many members of infinite families of simple
groups (see below). By 1972, only three new groups were added to this list. M.
Suzuki discovered the simple group with order 29,120 in 1960 as he discovered
the infinite family, Sz(2n). Z. Janko uncovered the simple group of order
175,560 in 1966, however this group was not a member of an infinite family
(that is, it is a sporadic simple group). In 1967, Hall and Janko discovered
a simple group (J2) with order 604,800 which was also sporadic. None of
these three groups was discovered because of work done on the range problem.
Apart from these three, by 1900 those simple groups with orders less than
one million were generally known to be simple before they were encountered
in the course of the range problem. They consist of classical linear groups,
alternating groups, and the Mathieu groups.
18
2.3 Some Examples
As examples of what the earlier work on the range problem was like,
I have examined groups of various orders to demonstrate that they are not
simple below.
Easy violation of Sylows third theorem (2.4), and use of Corollary
(2.8):
Example 1 If G = 54,587 = 132 17 19, then G is not simple:
We only need to look at the possible number of Sylow 13subgroups to
show that there is only one, thus it must be normal by (2.8). Note that by
Lemma (2.9) the number of Sylow psubgroups must divide the remaining
numbers left in the order of the group. Thus we have ni3 = 1,17,19, or
17.19 (= 323). Only 1 = lmod 13, thus n i3 = 1.
Example 2 If G = 35,321 = 11 132 19, then G is not simple:
This works in the same manner as above; we shall look for the number of
Sylow 11subgroups to show that there can only be one:
tin = 1,13,132,19,13 19, or 132 19.
If we check each, none except 1 is = 1 mod 11. If we had looked first at
rii3, we would have found that ni3 could be 11 19 which is = 1 mod 11.
Example 3 If (7 = 7480 = 23 5 11 17, then G is not simple:
This is an even order that works in the same manner. Notice the large
number of possibilities for nyj :
n17 = 1,2,4,8,5,11,10,22,20,44,40,88,110,220, or 440.
19
However, none of these are = 1 mod 17, thus the Sylow 17subgroup is
solitary and normal.
Easy violation of Lemma (2.9)
Example 4 If G = 7260 = 22 3 5 ll2, then G is not simple:
The possibilities for rin are 1,2,4,3,5,6,12,10,20,30, and 60.
If we ignore 1 for the moment, we can exclude all possibilities except 12
by (2.4) So if we assume G is simple, then nn 12. But notice that
G does not divide 12! since there is no second factor of 11 in 12!. Thus,
by (2.9), we have a contradiction aiid G is not simple.
Example 5 If G = 6468 = 22 3 72 11, then G is not simple:
The possibilities for nn, excluding the smallest factors since they cannot
be = 1 mod 11, are: 1,12,14,28,21,49,147,98,196,294, and 588. All
except 1 and 12 violate (2.4). Thus, if we assume G is simple, then
nn = 12. Once again, G does not divide 12! since there is no second
factor of 7 in 12!. Thus, by (2.9), we have a contradiction and G is not
simple.
Notice that for (2.9) to work, np must be fairly small. Here are a
couple of examples where np is too large to use (2.9), and a different technique
is needed: counting elements.
Example 6 If G = 616 = 23 7 11, then G is not simple:
Assume that G is simple. The possibilities for nn are the following:
nn = 2,4,8,7,14,28,56. Notice that only 56 = 1 mod 11, so we may
20
rule out the other possibilities. Can we also rule out 56 using (2.9)? No,
56 is large enough that G  56!. Let us check n7 for an easier approach.
n7 2,4,8,11,22,44,88. The only possibility that does not violate (2.4)
is 22, and similarly, G  22! since 22 is large enough. Thus, we have
nn = 56 and n7 = 22. A new strategy is needed for this problem. We
know 5'y/n(G)and \Syl7(G) \ and we know that each Sylow 11 and Sylow
7subgroups have 11 and 7 elements in them respectively. Any group of
prime order is also cyclic and we know that two different cyclic groups of
the same order that have more than one element in common must be equal.
Thus, each of the elements of Syln(G) and Syl7{G) must intersect only
trivially. We could count the elements in each. We have 56 groups with
11 1 distinct elements in each. The number of elements in Syln(G) is
then 56(11 1), and similarly, the number of distinct elements in Syl7(G)
is 22(71). We have accounted for 56(11 1)+22(71) = 560+132 = 692
elements so far. There are only 616 elements in the group, so we have a
contradiction. Thus, our assumption was incorrect, and G is not simple.
Example 7 If G = 520 = 23 5 13, then G is not simple:
This is similar to the above order. Assume that G is simple. Note the
possibilities: nn = 2,4,8,5,10,20,40 and n5 = 2,4,8,13, 26,104,52. Us
ing (2.4), we find that n13 = 40 and n5 = 26, and both numbers are
too large to use (2.9). Noticing that the subgroups in Sylu(G) and
Syls(G) are of prime order, thus cyclic, we may count elements. We
have 40(13 1) + 26(5 1) = 480 + 140 = 584 > 520. Thus, we have a
21
contradiction, and G is not simple.
The following two are more difficult cases using (2.9) and (2.4).
Example 8 If G = 800 = 25 52, then G is not simple.
Assume that G is simple. Notice that n5 = 2,4,8,16,32 and only 16 =
1 mod 5. Also, G  16!. So n5 = 16. Notice also that 16 is not =
1 mod 52 so we may use (2.4), the later half, which states that there
exists S and T e Syl$(G) such that S / T and 52 > I# : SflTl by
contrapositive. This implies that \S : S fl T\ = 5. (This is because S fl T
is a subgroup of S and \S\ = 52 thus if S' fl T ^ {1} which is necessary
if 52 > \S : S fl T, then S fl T\ must be 5 or 52. It cannot be 52 because
that would imply that S fl T = S = T) By Lagranges theorem, (2.1), we
have that \S\ = 52 = \S : S D T\ 15 fl T\ = 5 5 Thus, \S fl T\ = 5 and we
may use Corollary (2.12) which states that since 5 is the smallest prime
divisor of \S\, S fl T < S and by the same argument, S fl T < T. Consider
the normalizer of S fl T in G, Nq(S fl T). By the previous discovery, we
have that S C Nq{S fl T) and also T C Na(S fl T). Thus S and T must
be subgroups in the Syl5 subgroup of NG{S fl T). Since NG(S D T) C G,
by Lagranges theorem again,
\Na{SnT)\ = S \Ng(S n T) : S\ = 52 \NG(S n T) : S\.
Thus, \Ng(S DT) :S\ has to be = 2,22,23,24, or 25. If we look at the
number of Sylow 5subgroups in NG(S fl T), we see that it must also be
2,22,23,24, or 25, depending on \NG(S fl T) : S\. One further condition,
that n5(NG(SnT)) is = 1 mod 5, leaves us with n5(NG(Sr\T)) = 24. This
22
implies that 24 divides \Ng(S D T). Thus, \NG(S n T) = 52 24 or 52 25.
If \Ng(S fl T)  = 52 25, then NG(S fl T) G, and S fl T <3 G which is a
contradiction to our assumption that G is simple. Thus, NG(S D T) =
52 24. But this implies by Lagrange that G : NG(S flT) = 2. Note
that G does not divide 2!, (or alternately, any subgroup with index
2 is normal). Thus, we have by theorem (2.11) that G is not simple,
a contradiction to our assumption, but the last alternative. Thus, our
assumption was incorrect, and G is not simple.
Example 9 If G = 864 = 25 33, then G is not simple.
Assume that G is simple. The possibilities for n3 are the following:
n3 = 2,4,8,16, or 32.
(2.4) eliminates all but 4 and 16. Using (2.9) and noting that G cannot
divide 4!, we are left with n3 = 16. But 16 is not = 1 mod 32, so we can
conclude by (2.4) that there exists S and T G Sylz(G) such that S
and 32 > \S : S D T\. Using the same process as above, we can conclude
that IS1: S fl T\ = 3 and by Lagrange, I# fl T\ = 32. By (12), since 3 is the
smallest prime divisor of l^l, Sf\T
Ng(SC\T) and T C NG(SC\T). We have by Lagrange that \NG(S fl T) =
 S'I \Ng(S fl T) : S\ = 33 I Na(SnT) : S\. And since NG{S n T) C G,
\Ng(S n T) : S\ 2,22,23,24, or 25. We know that n3(NG(S fl T)) must
divide \NG(S DT) : 51 and also that ns(NG(S fl T)) = 1 mod 3, thus
n3(NG{SnT)) = 22 or 24. If n3(NG{SnT)) = 24, then JVG(5nT) : S\ =
24 or 25, and JVc(5nT) = 33 24 or 33 25. \NG(S n T) cannot be
23
33 25, since that would make G = NG(S fl T) and thus not simple.
Suppose JVG(SnT) = 33 24. Then G : NG(SnT)\ = 2 and since \G\
does not divide 2!, G is not simple by (2.11). This is a contradiction
to our assumption, thus ./VG(SnT) 7^ 33 24. If ns(NG(S fl T)) =
22, then \NG(SnT)\ = 33 22,33 23,33 24, or 33 25. We know that
ATG(SnT) / 33 24, or 33 25. Thus, suppose A^G(5nT) = 33 23.
Then G: NG(S (~)T)\ = 22, and G does not divide 4!, showing that
G cannot be simple (2.11). Suppose that iVG(SnT) = 33 22. Then
\G : Nq(S fl T) = 23, and still the index is too small, and G / 8!. Thus
since this is our last alternative, we conclude that our assumption was
incorrect, and G is not simple.
The following example uses a well known theorem, The N/C Theo
rem (2.7).
Example 10 If G = 792 = 23 32 11, then G is not simple.
Assume that G is simple. The possibilities for nn are the following:
nn = 2,4,8,3,9,6,12,24,18,36, or 72. Only 12 = 1 mod 11, thus nn =
12. Look at one subgroup in Syln(G), say S Â£ Sylu(G). Let N be the
normalizer in G of S', N = NG(S). Then since nn = 12 = \G : N\ (2.10),
we know that AT = 2 3 11 by Lagrange. Let C be the centralizer of S in
G, C = Ca{S). We know by the N/C theorem that the factor group N/C
is isomorphic to a subgroup of Aut(S). The set of automorphisms of S has
order
10. By Lagrange again, since \N\ = 2 3 11 = \C\ iV : C\, the only choice
24
for \N : C is 2, thus C =311. We see that the centralizer in G of S has
Sylow 3subgroups. Let P G Syl^C). Then P = 3. Consider Nq(P) and
note that Ng(P) cannot equal G since we are assuming G is simple, and
P
of S, then S C CG(P). But CG(P) C NG(P), so S C Na(P) which means
that 1Vg(P) is divisible by 11. By (2.3), there exists a Q G Syls(G) such
that P CQ. But \Q\ = 32, so \Q : P\ = 3. By (2.10) then, P < Q. Thus,
Q ^ Ng(P), which implies that 1VG(P) is also divisible by 32. So the
least order of Nq(P) is 11 32, which means that G : Ng(P)\ < 23. But
23 itself is too small, since G cannot divide 23! This implies that G is
not a simple group (2.11), which is a contradiction, thus our assumption
was incorrect and G is not a simple group.
Notice that the strategy in the previous problem was to find a sub
group of G which has order large enough to make the index of it in G too
small to be divisible by the order of G, thus utilizing the theorem (2.11). The
way to find a subgroup of G large enough to achieve this is to examine cen
tralizers and normalizers of subgroups within G. The following example also
uses normalizers in conjunction with (2.10), digging a few layers deep into the
structure of the group.
Example 11 If G = 3465 = 32 5 7 11, then G is not simple.
Assume that G is simple. The following lists the possibilities for all
25
Sylp(G) subgroups:
rin = 3,9,5,7,15,21,45,63,105,315
n5 = 3,9,7,11,21,33,77,63,99,231,693
n7 = 3,9,5,11,15,45,33,99,55,165,495
n3 = 5,7,11,35,55,77, 385
The numbers in bold are those that do not violate either (2.4) or (2.9).
These numbers indicate that the only possibilities for iVG(sp), where
sp G Sylp, by (2.10) are the following:
JVG(5ll)=7ll
JVg(s5) = 3511, or 35
\Ng(s7)\ = 3711, or 57
iVG(S3) = 327, or 32
Working systematically, we shall try to eliminate each of these as possi
bilities. Suppose that .JVG(s5) = 3 5 11. Look at \Sylu\ in NG(s5),
denoted nn(WG(s5)) : nn(NG(s5)) = 1,3,5, or 15. Note that the only
choice that does not violate (2.4) is nn(NG(s5)) = 1. Thus, by (2.10),
1 = Ng(s5) : NNg{ss)(su(Ng(s5))\ and thus JVG(s5) = 3 5 11 =
NNg{s5)(su(Ng(s5)) by Lagrange. But NNG{s5)(sn(NG(s5)) is the nor
malizer in NG(s5) of a Sylow 11subgroup, and note that NG(sn) is the
group of all elements in G that normalize a Sylow 11subgroup. Thus,
NnMMNg(s5)) C NG(sn), which implies that 35 divides A^G(sn).
But we know that iVG(sii) = 711 from above, thus we have a contra
diction. We now know that jVG(s5) = 35, and n5 = 231. Suppose that
iVG(s7) = 3 7 11. Note that nn(NG(si)) = 1,3,7, or 21. By (2.4),
26
nn{NG(s7)) = 1 is the only possibility. Then by (2.10) and Lagrange,
Nng(S7)(su{Ng{s7)) =37ll. But this implies that 3  /VG(sn) = 711,
which is a contradiction. Thus, Ag(s7) = 57 and n7 = 99. Now look
at the possibilities for n^Na^Sy)): 1 or 7. By (4), ns(iVG(s7)) = 7, and
by the same argument as above, this implies that 7  Ng(s^). We have
from above that iVG(s5) = 35, thus we have a contradiction. The only
possibility is that one of the Sylow subgroups is unique, thus normal.
Therefore, G is not simple.
The strategy of this last example is to use theorems about the size
of Sylp(G) more than once to draw a contradiction. The following example
starts in this manner, then requires a method previously seen, and comes to a
conclusion with the same method used at first.
Example 12 If G = 760 = 23 5 19, then G is not simple.
Assume that G is simple. The following list the possibilities for the sizes
of all Sylp{G):
n2 = 5,19,95
n5 = 2,4,8,38,76,152,19
ni9 = 2,4,8,5,10,20,40
The numbers in bold indicate those that do not violate (2.4) or (2.9). The
following are the possible orders of the normalizers of the Sylow subgroups
27
by (2.10):
iVG(S2) = 235, or 23
JVg(s5) = 25
jVG(s19) = 219
We would like to determine /VG(s2), so suppose 1Vg(s2) = 23 5. Then
n5(NG{s2)) 1,2,4, or 8. We conclude by (2.4) that n5(NG(s2)) = 1.
Thus, using the same process as above, by (2.10) and Lagrange, we can
conclude that iWrG(S2)(s5(iVG(s2)) = 23 5. Since NNg^S2)(s5(Ng(s2)) C
Ng(s5), then 23 divides A^G(s5) = 2;5, a contradiction. Thus, jVG(s2) =
23, and n2 95. Note that 95 ^ 1 mod 22, so by (2.4), there exist S and
T G Syl2(G) such that S / T and 22 > \S : S D T\. This implies that
\S : S D T\ = 2. By (2.10) we have that
S'nT
In fact, our possibilities for A^G(5 flT) are: 23 19,23 5, or 23 5 19.
We may rule out \NG(S DT) = 23 5 19 since that would imply that
Ng(S fl T) G, and thus S n T < G, which is a contradiction. Suppose
that \Ng(SDT)\ = 2319. Then \G : NG(S nT)\ = 5, but G /5!, which
implies a contradiction by (2.9). Thus, NG(S fl T) = 23 5. Look at the
size of Syl5(NG(S flT)): = 1,2,4, or 8. By (2.4), n5 = 1. By (2.10)
and Lagrange, we have that NNG(SnT)(s5{NG(S fl T)) = 23 5 But this
implies that 23  ATG(s5) since NNa(SnT){s5(NG(S flT)) C A^G(s5), and
this is a contradiction since JVG(s5) = 25. Thus, our original assumption
must be incorrect, and G is not simple.
28
The orders used for these examples are obviously fairly small. As
one can guess, the larger the order, the more cumbersome are the choices for
such numbers as np(G). Take the simple group, J\ for example. This group
(described further below) has order 23 3 5 7 11 19. In order to determine
n19, one must consider 56 possiblilities. Out of this 56, there are four numbers
which cannot be eliminated using (2.4) or (2.9). Since 19 is the largest prime
divisor of G, nig should be the most accessible of all sizes of the Sylp(G)
to find. Imagine what the others must be like! The shear magnitude of the
problems increase as the orders become very large. Not all groups of large
order are difficult to handle, however. Take for example G = 1,000,000.
It is a simple matter of using (2.9) on the possibilities for n5 that proves G
is not simple. Nonetheless, when the larger orders are difficult, they can be
very difficult. They are generally more cumbersome when their orders are
comprised of quite a few primes close in size. It is no wonder that Marshall
Hall, Jr. employed computer assistance in the course of his completion of the
range problem up to order one million.
29
3. The Simple Groups
3.1 Infinite Families of Simple Groups
3.1.1 The Alternating Groups
I have often in my life ventured to advance propositions of which I was
uncertain; ... it is too much to my interest not to deceive myself that I
have been suspect of announcing theorems of which I had not the complete
determination ... subsequently there will be, I hope, some people who will
find it to their profit to decipher all this mess. (Galois [15])
The history of group theory itself begins with the discovery of the first
compositely ordered simple group, A5. The process that led to the discovery of
this simple group actually led to the idea of the study of group theory. It began
with Evariste Galois (18111832) who led a very short but mathematically
productive life, although it took time and scrutiny for anyone to understand
his ideas. The above quotation was on the final page written by Galois before
he died for so trivial a thing [16] in a duel when he was twenty one years
of age. Many of the terms that he used were not rigorously defined, and
his results were not often proven, being hurriedly jotted on a piece of paper.
Yet Galois did have the first concept of groups as we define them today, and
used them somewhat abstractly in his studies of solvable polynomials. Galois
was working on the popular algebra problem of the eighteenth and into the
nineteenth centuries, the factorability of polynomials over a field F.
30
Galois approach to this problem is rooted in the workings of permu
tations. The possible roots of a polynomial of degree n can be permuted in
n! different ways. For example, look at the fourth order polynomial in the
complex field: f(x) = (x2 + l)(x2 3). The four roots of the polynomial are
x = i, i, \/3, and y/%. Suppose we let
a = i
p=i
j = V3
6=y/3.
( a A
a p 7 o
Then we have permutations of these four letters such as R\ =
y (3 a 7 5
which switches a and f3 and leaves the other two fixed. There are 4! = 24
similar permutations. A subgroup of the group of 24 permutations can be
formed in the following way. Look at any polynomial equations involving a, /?,
7, or S. Some equations express a true statement if the numerical values of a, /?,
7, or 5 are substituted, and some do not. For example, the equation 72 3 = 0
is true for 7 = \/3, as is a+p = 0 for the given values of a and j3. An equation
such as 2(3 5 = 2 is obviously not true. The group of permutations which
preserve the truth of the true equations form a subgroup of the permutation
group. Notice that any true equation remains true if a. and /? are interchanged,
and similarly if 7 and 5 are interchanged. Galois called this subgroup of
permutations the group of the equation, G. In our example, this group consists
31
of
i2i
f a A
a p 7 o
$ j
, R2
( a A
a p 7 0
(3 5 7 j
, R3
( R A
a p 70
f3 a 5 7
,/ =
(a A
a p 7 (3
Ka (3j S }
The first concept of a normal subgoup was born by examining the
group G. Choose a polynomial expression, T, which is rational in the roots
of our original equation but has the following property: its numerical value,
t, stays fixed for some elements of G, but changes for others. Then those
elements of G which fix t form a subgroup, H, of G. Galois showed that if t is
a root of the (irreducible over F) binomial equation xpc = 0 where p is prime,
then the subgroup H is in fact normal in G. This process continues to reveal a
method of solving equations by radicals, and also the inspiration for studying
simple groups. Form a new field F(t) which is the smallest field containing
both F and t. The subgroup H is then the group of the equation over the
new field, F(t). Repeat the above process on H to find a normal subgroup
of H, and a new field, F(t,t\) where t\ is the numerical value of the chosen
expression. The process can be repeated until we are left with the identity
permutation as the subgroup. In this case, the original equation is said to
be solvable by radicals over the created field, F(t,ti, ...,tn). Furthermore, we
have a series of normal subgroups much like
1 Ho <1 i?i<] ... <] Hn1 <3 Hn G
where the index of one in the other, was shown to be the prime
number p in the appropriate equation xpc 0. This looks remarkably like the
composition series discussed earlier, and since each index is prime we see that
each composition factor, \Hi/Hii \ must be trivially simple. Galois discovered
32
that an equation was solvable if each index in the composition series was prime,
and not solvable if some index was not prime. This is precisely what happens
to quintic equations. Some composition factor in the composition series is
compositely simple, not having prime order, and the end result of the identity
permutation is never obtained.
The simple group that was discovered by Galois by way of the in
solvability of the quintic was the simple group of order 60. By 1832, Galois
recognized this group as simple stating The smallest group of permutations
which an indecomposable group can have, when this number is not prime
is 5 4 3 [15]. Galois stated this without proof and it wasnt until 1870
that Jordan would verify this result. In fact, Jordan gave better definition to
the notion of a composition series which was only one great feat of his 1870
work, Traite des substitutions et des equations algebriques [15], which further
inspired the study of simple groups. By this time, mathematicians were still
concerned with the solution of algebraic equations, and this was the foremost
purpose of the Traite. The use of permutation groups was still being explored
and expanded, and groups were generally represented as such. Thus, Jordan
discovered that the simple group of order 60 which was tied to the quintic
equation was actually the alternating group on five letters, A5. An alternating
group is the subgroup of the permutation group made up of all even permu
tations. (A permutation is even if it can be written as a product of an even
number of 2cycles, or transpositions.) Jordan went further than proving the
simplicity of A5. He presented a (flawed) proof for the simplicity of all alter
nating groups, An, for n > 5. This was the first infinite family of simple groups
33
to be discovered. As an example of the permutation group theory used, the
following is a proof for the simplicity of A5:
Theorem 3.1 A5 is simple.
Proof: The cycle structures of the elements in A5 are the following: 15,1 22,
l2 3, and 5. This notation indicates that there are permutations which fix
five letters (the identity), fix one letter and has two 2cycles, fix 2 letters and
has one 3cycle, and which has one 5cycle. The orders of the elements in A5
which are made up of these cycle structures can be obtained by finding the
least common multiple of the sizes of cycles for each type. That is, the order
of the elements that are made up of two 2cycles and fix one point is 2 (LCM
of 2 and 1), etc. as shown below:
s structure order of elements number of elements
l5 1 1
1 22 2 15
l2 3 3 20
5 5 24.
The last column above shows the number of elements of each order.
These numbers are easily obtained by looking at the order of A5. For example,
the number of elements of order 3 is the number of elements of order
5 is etc. To show that A5 is simple, we shall proceed by contradiction.
Suppose that A5 contains a normal subgroup, S, which is not the identity
or A5 itself. The possible orders of S must divide 22 3 5. Suppose that
3  S\. Then S contains a Sylow 3subgroup of A5 and since S is normal and
every two Sylow psubgroups are conjugate (2.14), S must contain all Sylow
34
3subgroups. Thus, S contains all elements of order 3. There are 20 elements
of order 3, so 5 > 20 (accounting for the identity). Also, 3  l^l and \S\  A5,
so \S\ = 30.
Now suppose that 5  <5. By the same argument as above, S contains
all Sylow 5subgroups, and thus all 24 elements of order 5. So \S\ > 24, thus
 S  = 30. Since 30 is divisible by both 3 and 5, S must contain all elements of
both orders, 20 + 24, but this is impossible if l^l = 30.
So suppose S'! = 4. Then S would be a normal Sylow 4subgroup,
and thus would be the unique Sylow 4subgroup. But there are 15 elements of
order 2, so this is also impossible.
Finally, suppose l^l = 2. Then \Aut(S)\ = 1 since cp(2) = 1. (2.13)
Using the N/C theorem (7), Na5{S)/Ca5{S) = 1 thus, Nas(S) = Cas(S).
Since S is normal in A5, Na5{S) = A5 which implies that Ca5{S) = A5. This
is not true, since a counterexample can be found easily as a 3cycle which does
not commute with a product of two 2cycles. So none of the possibilities work,
and our assumption must be incorrect. Therefore, A5 is simple.
3.1.2 Simple Groups of Lie Type
The remainder of the infinite families of simple groups can be clas
sified as Lie groups. These include the classical groups, the groups of type
G2, the Chevalley groups (of types E,Ee,E7, and E$), the twisted groups (of
types E6 and ZU), the Suzuki groups and the Ree groups (of types G2 and
F4). These groups arise as automorphism groups of corresponding simple Lie
algebras. In general, since the theory of Lie algebras is too extensive for this
35
paper, a Lie algebra is a vector space over a field with a product [X, Y] that
is linear in both variables which also meets the following criteria:
1) [X, X] = 0 for all X in the vector space.
2) [[X, Y],Z] + [[Y, X], X] + [[Z, X],Y] = 0. (the Jacobi identity)
3.1.2.1 The Classical Linear Groups
It was Jordan again in his Traite who found the next four infinite fam
ilies of simple groups, although he was not completely aware of the simplicity
of each. Jordan obtained orders, generators and the factors of composition of
some of these groups and was not explicit about the infinite families involved.
We have seen how the simplicity of the infinite family PSL(m,pn) was finally
proven by Dickson in 1897. In fact, Dickson worked on extending Jordans
results on all of the linear groups from 1897 to 1899. Dickson and Dieudonne
are also credited with further investigating all of the linear groups in the years
1948 to 1958. The groups are now known as the projective special linear,
the symplectic, the orthogonal, and the unitary groups. All four are collec
tively called the classical linear groups. They are each groups of matrices.
The construction of the first two are given below, and the construction of the
orthogonal and unitary are similar in that they are each groups of invertible
matrices factored out by the groups center.
Projective special linear: The general linear group, GLn(q) is the
group of all nonsingular linear operators of a vector space V where V has
dimension n over the field of order q. Thus, GLn(q) is a group of n by n
matrices. The order of GLn(q) can be given by the following:
\GLn(q)\ = (q l^WV 1)...(9 1).
36
The subgroup of matrices with determinant 1 is normal and called the special
linear group, SLn(q). The order of SLn(q) is given by q (2" ) (q2 1 )...{qn 1).
The center, Z, of GLn{q) consists of transformations of the form Tx = Xx
for A not 0. The center of SLn(q) can be denoted Z n SLn(q) and the factor
group is the projective special linear group, PSLn(q). Its order is given by
the following \PSLn(q)\ qn('n~1^2(q2 l)...(<7n 1) Let the field be
the Galois field GF(q) where q is a power of a prime. This group is simple for
n > 2 except for PSL2(2) and PSL2(3).
Let us look at a specific example of a projective special linear group.
The simple group PSL3(2) is isomorphic to PSL2(7), both with order 168.
We would construct PSL3(2) by looking first at GL?J(2) which consists of
all nonsingular 3 by 3 matrices over the Galois field, GF(2). For example,
the matrix
1 1 0
10 1
is an element in GLS(2). The order of GL3(2) is
Oil
(2 1)23(23 1)(22 1) = 168. Note that this is the same as the order
of PÂ£L3(2) and indeed, they are isomorphic. The reason for this is that all
matrices in GL3(2) have determinant equal to 1 mod 2, thus all elements in
GL3(2) are also in 5X3(2). The center of SL3(2) consists of the identity only,
thus SL3(2)/ZnSL3{2) = SL3(2) = PSL3{2).
The construction of PSL2(7) isomorphic to PSL3(2) begins with
GL2(7), the group of nonsingular 2 by 2 matrices over GF(7). GL2(7) has
order 6 7 (72 1) = 2016. An element in this group looks something like
37
6 2
1 3
or
4 1
0 5
where the matrix entries are modulo 7. If we restrict
ourselves to all matrices in GL2{7) with determinant 1 mod 7, for example
33 ,
, we have SX2(7) with order 336. The center of GL2(7) consists of
1 6
1 0 2 0 6 0
matrices like , . The center of 6X2(7) are those
1 0
0 1
1 0 1 to 0 6 0
0 1 J 0 2 0 6
matrices in the center with determinant 1 mod 7, which are only 2,
. The simple group PSL2(7) is the factor group of SL2(7) and
these two elements. Comparing the order with the formula given above, we
see that \PSL2(7)\ = Â§ 7 (72 1) = 168.
Projective Symplectic: Suppose that the vector space V from above
has a skewsymmetric, bilinear nonsingular scalar product so that (x,y) =
(y,x), and (x,x) 0. The symplectic group, Spn(q) where n = 2m, consists
of those linear transformations which preserve the above symplectic form. In
particular, if A, B, C, and D are m x m matrices, then the transformation
and
6 0
0 6
represented by the matrix
A B
is symplectic exactly when the following
C D
hold: A*C ClA = 0, A*D CtB = I, and B*D DtB = 0 [5]. The
projective symplectic group, PSpn(q), is the factor group Spn(q)/Z(Spn(q))
where Z(Spn(q)), the center of Spn(q) is made up of scalar matrices. PSpn(q)
is simple except for PSp2{2), PSp2{3), and PSp^(2). The order of PSpn{q) is
given by the following formula: (qm2(q2m 1 )(q2m~2 l)...(q2 l))/(q 1, 2).
38
An example of a projective symplectic group is PSp2{9) which con
tains 360 elements and is isomorphic to A6. Sp2(9) is a subgroup of GL2{9),
the set of 2 x 2 matrices over the Galois field of 9 elements. First, we construct
GF(9) by looking at the irreducible polynomial x2 + 1 over Z3. We find that
GF(9) = {ax + b+ (x2 + 1)} and the elements are
{0,1, 2, x, x + 1, x + 2, 2x, 2x + 1, 2x + 2} .
Following the equations above, and simplifying our example by only looking
at elements for which B = C = 0, we can write a couple of elements of 5^2(9):
X + 1 0 , and 1 to ss 0 . The two elements in Z(Sp2(9)) are 2 0
0 x + 2 0 x 0 2
and
1 0
0 1
, and since PSp2(9) is the factor group of Sp2(9) and these two
x + 1 0 2i 0
elements, we know that
0 x + 2
One can easily verify that (x + 1)(T + 2'
and are also in PSp2(9).
0 x
= 1, and (2x)(x) = 1 in GF(9).
3.1.2.2 Other Lie Groups
A brief mention of the history of other groups of Lie type is in order.
During the period 1901 to 1905 a new family of simple groups of Lie type was
discovered by Dickson. Until 1955, classical linear and this new family were
the only simple groups of Lie type known. Claude Chevalley produced a much
needed new way of approaching these simple groups and in the process, he
discovered several more infinite families of simple groups of Lie type. These
are referred to as the Chevalley groups. Chevalleys progress on the groups of
39
Lie type successfully increased the interest in the field, and it wasnt long before
new infinite families of simple groups of Lie type were found. In particular,
in 1960 Suzuki discovered his infinite family while working on what is now
called a classification problem (see Chapter 4). He was trying to find all
simple groups in which the centralizer of an involution (that is all elements
that commute with a particular element of order two) is a group of order 2n.
In the process of trying to eliminate all possibilities except for PSL(2, 2n)
and PSL(3,2n), n> 2, Suzuki found another family with the given property.
These are Sz(2n). In 1961, Rhimak Ree was analyzing the Suzuki groups using
a particular method (Steinbergs) which had produced infinite families of Lie
type before, and came up with two additional families. Thus, the Chevalley,
Steinberg, Suzuki, and Ree groups are the simple groups of Lie type along
with the classical linear groups.
3.2 The Sporadic Simple Groups
The remaining known simple groups do not fit into any large model
of similar attributes as do the infinite families. They were discovered often
one by one. Some do fit together by method of discovery or by construction.
We will examine these properties briefly below. First, the following table lists
the 26 sporadic simple groups, their order (if not too large), their discoverer
(according to some references), and the date of their discovery.
Name Order Discovered by Date
Mn 24 32 5 11 Mathieu 1895
M\2 26 33 511 Mathieu 1899
M22 27 32 5 7 11 Mathieu 1900
M23 27 32 5 7 11 23 Mathieu 1900
M2 4 210 33 5 7 11 23 Mathieu 1900
Jl 23 35 711 19 Janko 1966
J2(HaJ) 27 33 52 7 Hall,Janko 1967
MHJM) 27 35 5 17 19 Janko, Higman, McKay* 1969
HS 29 32 53 7 11 Higman,Sims 1968
McL 27 36 53 7 11 McLaughlin 1969
Suz 213 37 52 7 11 13 Suzuki 1969
He 210 33 52 73 17 Held,Higman,McKay 1969
Coi 221 39 54 72 11 13 23 Conway,Leech* 1969
Co2 218 36 53 7 11 23 Conway* 1969
C03 210 37 53 7 11 23 Conway* 1969
Fi 22 217 39 52 7 11 23 Fischer 1969
Fi 23 218 313 52 7 11 13 17 23 Fischer 1969
Fi 24 * Fischer 1969
Ly * Lyons, Sims 1971
Ru 214 33 53 7 13 29 Rudvalis, Conway, Wales 1972
O'N 29 34 5 73 11 19 31 0Nan,Sims 1973
M * Fischer 1974
B * Fischer 1974
f3 215 310 53 72 13 19 31 Thompson,Smith* 1974
f5 214 36 56 7 11 19 Fischer,Smith, Harada* 1974
h * Janko* 1975
Table 3.2: The Sporadic Simple Groups
* The names of discoverers followed by a star are those which have
some discrepancy depending on sources. Those orders denoted by a star are
41
too large to fit this table. For example, the order of the group M, the largest
of the sporadic simple groups is
808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000
3.2.1 The Mathieu Groups
The Mathieu groups, Mn, Mn, M22, M23, and M24 are the earliest
sporadic simple groups to be discovered. They were described by Emile Math
ieu (1835 1900) in 1861 and 1873. Mathieu was influenced by Cauchys work
on permutations. Mathieu was investigating multiply transitive functions, and
thus permutation groups and multiply transitive permutation groups. A per
mutation group on a set A is said to be ntransitive if for any ordered pair of
ntuples of elements of A, there exits some element of the permutation group
that maps one tuple to the other. That is, Xig = yi for 1 < i < n where Xi
and yi G A and g 6 permutation group of A. A transitive function is one
which is left invariant under the permutations of a transitive group, which
was discovered by Mathieu. In the course of his work, Mathieu attempted to
extend transitivity by constructing an ntransitive permutation group out of
a (n1)transitive permutation group. He was able to find an algorithm for the
construction of these groups, when their construction was possible. The high
est transitivity found in a simple group is 5transitive, and Mathieu discovered
the 5transitive permutation groups on 12 symbols and on 24 symbols which
are M12 and M24. The other Mathieu groups arose as subgroups of these and
a subgroup of M23. For example, Mn is the subgroup of M42 formed as the
42
stabalizer of a point in Mi2 Each of the Mathieu groups are multiply transi
tive. The simplicity and uniqueness of the Mathieu groups was not expressed
until the 1930s in a paper by Witt who was describing what is called the
Steiner system. The Mathieu groups are now normally described in terms of
this system.
3.2.2 Centralizer of Involution Problems
The next sporadic simple groups were not discovered until around
one hundred years after Mathieus find. The first of these is Jankos first, Ji,
and the method by which it was discovered became an important part of the
theory of simple groups and an important method to discover other simple
groups. The central feature of the method is the centalizer of an involution,
or the centralizer of an element of order two. We have seen how centralizer
of involution questions led to Suzukis infinite families of simple groups of
Lie type. The centralizer of an involution as an entity is important due to a
number of results. Two of these are a theorem due to Brauer and Fowler and
the FeitThompson Theorem (or Odd Order theorem), which are both looked
at below.
Because an involution is an element of order two, the order of a group
containing an involution must be divisible by two. If it were guaranteed that
a simple group contained an involution, this may increase the potential of
classifying simple groups according to something related to involutions. This
result was indeed obtained, in the FeitThompson Theorem. It was not a swift
theorem to come up with however, and the odd versus even order of simple
43
groups was a long standing question. In fact, conjectures on this question date
back to 1895 and Burnside. Burnside had a good hunch that simple groups
must necessarily have even order, and from 1895 to 1901, he attempted to show
this. He was successful at proving that all simple groups with orders under
40,000 had even orders, yet he could not generalize his result. He believed that
the necessary technique to prove his conjecture was character theory. The
problem came alive again in 1957 with the work of Suzuki who was indeed
using character theory. Suzuki was able to prove that any simple group in
which the centralizer of any element (other than the identity) was abelian has
even order. This result was extended in 1960 by Feit, Hall, and Thompson
who proved that a simple group must have even order if the centralizer of any
nonidentity element is nilpotent, i.e., all of its Sylow subgroups are normal.
Three years later, the same two, Feit and Thompson, took 255 pages of the
Pacific Journal of Mathematics [6] to prove that all groups of odd order are
solvable. This means that the composition series of a group of odd order
contains composition factors of prime order, which indicates that the group
is not simple. Thus, any simple group must have even order, and therefore,
must contain involutions.
A result pertaining directly to the centralizers of involutions was
actually found earlier than the FeitThompson Theorem. In 1954, Brauer
and Fowler proved that there are at most a finite number of simple groups in
which the centralizer of an involution has a given structure:
Theorem 3.2 IfG is a finite simple group of even order, andt is an involution
in G, then G < (CGr(i)2)!.
44
Ca(t) denotes the centralizer in G of t. Since there can only be a finite
number of groups with orders less than a particular number, then there are
only a finite number of groups with the centralizer of an involution isomorphic
to a given centralizer. This provided the idea of at least some classification
of finite simple groups by the structure of the centralizer of involutions. The
importance of this result was furthered by the FeitThompson Theorem since
then the result pertained to all simple groups, not just simple groups of even
order. This theorem has been improved upon in the more recent years in many
variations using the idea of a central involution which is an involution in the
center of a Sylow 2subgroup. In general, it has been established that if a
centralizer of a central involution in a questionable simple group is isomorphic
to the centralizer of a central involution in a known simple group, then the two
simple groups are isomorphic. These are powerful results which may allow for
the characterization of a simple group by its centralizer of a central involution.
An example of this type of theorem is the following due to Brauer:
Theorem 3.3 Let G be a simple group which contains an involution whose
centralizer is isomorphic to GL2(q) factored, by a subgroup of odd order in the
center of GL2(q), and where q is an odd prime power congruent to 1 mod 4.
Then either
1 .G^PSL3(q), or
2. G = Mu and q = 3.
There are many other such theorems, and the theory involved in the study of
centralizers of involutions is extensive. This paper will only be able to concern
itself with a brief description of the discovery of some of the sporadic simple
45
groups due to centralizer of involution theory.
Let us return to the next sporadic simple group to be discovered, J\.
The story of Jankos first group begins with the centralizer of the involutions
in one family of Ree groups of Lie type, denoted R(3n). It was found that the
centralizer of an involution of R(3n) is isomorphic to the group Z2 x PSL2(3n),
the external direct product. It was also noted that the Sylow 2subgroups are
elementary abelian of order 8. Thus, an interesting task became to determine
all simple groups with Sylow 2subgroups with the above properties which have
centralizers of involutions isomorphic to Z2 x PSL2(pn), p an odd prime. For
pn = 5, the new simple group J\ was discovered. Janko proved the following
theorem:
Theorem 3.4 If G is a simple group with abelian Sylow 2subgroups of order
8 and the centralizer of an involution of G is isomorphic to Z2 x PSL2(3),
then G is a uniquely determined simple group of order 175,560. Moreover,
Gis isomorphic to the subgroup of GL7(11) generated by the following two
elements of order 7 and 5:
0 1 0 0 0 0 0 3 2 1 1 3 1 3
0 0 1 0 0 0 0 2 1 1 3 1 3 3
0 0 0 1 0 0 0 1 1 3 1 3 3 2
0 0 0 0 1 0 0 and S2 1 3 1 3 3 2 1
0 0 0 0 0 1 0 3 1 3 3 2 1 1
0 0 0 0 0 0 1 1 3 3 2 1 1 3
1 0 0 0 0 0 0 3 3 2 1 1 3 1
Janko was the lucky receptor of further inspiration which led to two
46
other sporadic simple groups, J2 and J3. After the discovery of Ji, Janko
looked further for possible centralizers of involutions inspired by those in the
Mathieu groups. He tried a centralizer of an involution which was isomorphic
to the extension of a group of order 32 by A5. He actually found two new
groups with the same centralizer of an involution, J2 and J3. Hall and Wales
proved the existence of J2, and Higman and McKay proved the existence of
Js
The question of the existence of two simple groups with isomorphic
centralizers of involutions led to the discovery of the next sporadic simple
group in the story. (We now cease chronological order). D. Held knew that
the groups M24 and PSL5(2) have involutions with isomorphic centralizers.
While investigating this phenomena, Held discovered yet another simple group
with the same centralizer of an involution, He. This is the only case of three
simple groups with isomorphic centralizers of involutions.
The next sporadic simple group to be obtained by examining cen
tralizers of involutions is Ly. John McLaughlins group, Me, to be discussed
below, has a centralizer of an involution which is isomorphic to the group A8
which denotes the perfect extension of A8 by Z2. The idea then arose to study
centralizers of involutions which are isomorphic to An for n > 5. On such
an investigation, Richard Lyons, who was a student of Thompsons, made the
following discovery:
Theorem 3.5 If G is a simple group in which the centralizer of an involution
is isomorphic to An, n = 10 or 11, then n = 11, and
\G\ = 28 37 56 7 11 31 37 67.
47
In fact, the result was shown that simple groups could only arise from central
izers of involutions isomorphic to As and An. Incidentally, It was Janko who
had worked on this problem. He showed that when n = 9 and 10, there were
no simple groups with the said centralizer of an involution. He gave up before
working on n = 11. He did however discover the last sporadic simple group
falling under the category of centralizer of involution problems, and that was
J4.
3.2.3 Rank 3 Permutation Groups
The group J2 has a structure which became important to the con
struction of four more sporadic simple groups. J2 is said to be a primitive rank
3 permutation group. A group G has permutation rank r if G is transitive on
a set 17 and the subgroup of G that fixes a point of Â£7 has exactly r orbits on
fh Recall that a group is transitive if for a set 17, and any two elements a and
(3 e 17, there exists an element g e G such that a g = p. Also, the subgroup
of G that fixes a point of 17 are those elements in G for which a g = a. The
orbits on 17 are sets of the form {a g \ g Â£ G} C 17. The orbits of 17 partition
17. The group J2 fits this description if one considers the maximal subgroup of
index 100, H C J2. The permutation representation of J2 on the right cosets
of H is a transitive action which produces a primitive permutation represen
tation of J2 of degree 100 (i.e., J2 is a transitive permutation group of degree
100), which takes the role of the set Q above. On this set, H fixes one point
and its action produces two other orbits, rendering J2 a rank 3 permutation
48
group. In fact, the existence of J2 was proven using the theory of rank 3 per
mutation groups. The maximal subgroup H happened to be isomorphic to the
simple projective special unitary group, Â£/3(3). Donald Higman and C. Sims
noted the similarity in permutation properties between the groups 173(3) and
M22, and in record time were able to construct a new simple primitive rank
3 permutation group using M22 as the maximal subgroup and extending it
to obtain the group HS. A similar technique was used by McLaughlin who
started with the group Â£4(3) to extend it to a rank 3 permutation group that
is simple, called McL. Suzuki obtained his sporadic simple group Su in the
same manner starting with G2(4) (a Chevalley Lie type simple group). Finally,
the fourth rank 3 permutation group was constructed by Rudvalis using the
Ree group, 2i4(2). It is Ru.
3.2.4 The Remaining Sporadic Simple Groups
This shall serve to briefly describe the discovery of the remaining
sporadic simple groups. The Conway groups, Coi, Co2, and Co3 came out
of the study of an automorphism group of a lattice called the Leech lattice
which is determined by a set of vectors in 24dimensional Euclidean space
with integral coordinates. The three simple groups happen to have been sub
groups of this automorphism group. The Fischer groups, Fi22, Fi23, and Fi'2A
were discovered by Fischer while studying classes of 3transpositions. These
are conjugacy classes generated by involutions such that the product of two
involutions in the class has order 2 or 3. Fischer generated groups by these
classes, and put further conditions on the groups proving that the new group
49
is either a symmetric group, a certain classical group, or one of the three Fis
cher groups. Fischer then turned to groups generated by {3,4}transpositions
(two involutions in a class have a product of order 2, 3, or 4). Two groups,
B and M, or Baby Monster and Monster were discovered. The Monster is
the largest sporadic simple group, and a representation for it was obtained by
hand by Robert Griess. It was in terms of square matrices that were 196,883
by 196,883 in size. The groups B, F3, and F5 are actually subgroups of the
Monster. F3 was found by Thompson, and F5 is attributed to Harada, Norton,
and Smith. The ONan group came out of the study of groups with particular
Sylow 2subgroup structure.
The methods used to discover new sporadic simple groups were often
haphazard, as Daniel Gorenstein says, some of the groups seemed literally
plucked from thin air [9]. Sometimes the techniques used were character
theorybased. In fact, Feit, Thompson, and Brauer were quite well known
for their work in and development of character theory. There are really three
phases in determining a new simple group, and only one of those phases I
have taken consideration of here. There is the discovery, which is what I have
described, there is the existence and there is the uniqueness. Often several
different individuals contribute to the determination of the existence and the
uniqueness of a new simple group. The discoverer is generally who the group is
named after. The simple groups found later than Ji had the timely advantage
of computers to aid in their discovery, existence, and uniqueness questions.
50
4. The Classification Theorem
4.1 History
The study of centralizers of involutions proved not only very useful
in locating certain sporadic groups, but also marks what some would consider
the start of the classification project. As noted above, in 1954 Brauer made
his great discovery that there are only a finite number of groups with their
centralizers of an involution having a particular structure. This seemed to spur
the idea of the characterization of simple groups according to their centralizer
of an involution. It was in fact Brauer who suggested such a thing, and was
successful with his use of character theory in certain cases. Others contributed
to this line of study, and some good results were obtained, often with the
discovery of sporadic simple groups. Brauers ideas served to provide a new
avenue down which some could dream of an overall classification of all finite
simple groups. There were also certain advances in theory that inspired many
to take part in the study of simple groups. The work of Brauer and Suzuki in
character theory provided one. The new discoveries about Lie groups in the
1950s is another. But in the 50s, there was still much to be accomplished
before a classification idea could become a reality.
The 1960s provided the study of simple groups with some of those
high powered results it needed. The most influential is the famous 1963 Feit
Thompson theorem, or the odd order theorem, which states that all groups of
51
odd order are solvable. It was not only the result that was terribly influential,
but also the structure of this 255 page proof. Thompson was also respon
sible for another very important result which took 410 pages and six years
(19681974) to complete. This is the classification of minimal simple groups,
or those simple groups which have only solvable groups as subgroups. Fol
lowing Brauers program, Suzuki was able to characterize all simple groups in
which the centralizer of an involution has a normal Sylow 2subgroup in 1965.
Sylow 2subgroups were becoming as telling as centralizers of involutions, and
many results stemmed from their study. In particular, Gorenstein and Walter
characterized simple groups with dihedral Sylow 2subgroups also in 1965. In
1969, Walters classified simple groups with abelian Sylow 2subgroups. These
are general characterizations. An example of a specific characterization is
Glaubermans Z*theorem of 1966 which showed that every involution is con
jugate to another involution in its centralizer. These are only a few of the
important steps taken in the 1960s, and many other results were to follow.
By the 1970s, there were many roads to classification, although no
systematic idea of its achievement. There were also many sporadic groups
discovered in the 1960s, and some wondered if there was an endless supply of
them. Thus, in 1972 at the University of Chicago, when Gorenstein presented
his idea of a 16step plan to classify all simple groups, not many were opti
mistic. Gorenstein projected that to complete his program would take about
thirty years. The task seemed daunting, yet a few tackled portions of the
plan. The project was propelled rather suddenly by a newcomer, Michael As
chbacher, who came on now like a whirlwind, moving directly to a leadership
52
position and sweeping aside all obstacles, as he proved one astonishing result
after another [9]. The results being made at this time were obviously highly
complex, and therefore cannot be handled in this paper. It should be noted
that the original plan of 30 years was decreased to an actual 10 years, and
Gorenstein attributes this to Aschbacher. The completion of the classification
theorem took place in January of 1981.
Some Methods
The methods used in pursuing the idea of the complete classification
of finite simple groups naturally changed as progress was made. As can be
noted from previous chapters, character theory was used frequently for many
results. It turns out that character theory is limited in handling large simple
groups. Smaller groups, such as lower ordered groups, or groups with Sylow
2subgroups that are restricted structurally (such as abelian) are good candi
dates for the use of character theory in examining them closely. However, as
the questions about the groups internal structure became more broad, new
techniques were needed. These techniques are called local grouptheoretic anal
ysis, or local analysis. It was the FeitThompson theorem that initiated the
practice of local analysis. The predecessors of the FeitThompson theorem,
Suzukis abelian centralizer result and the Feit, Thompson and Hall result on
nilpotent centralizers (see p.45) used character theory to develop the lattice
of proper subgroups of the group in question. This required analysis of ev
ery subgroup. This process could not be used in the FeitThompson theorem,
since there was no information on the structure of centralizers to rely on. A
new set of techniques was developed by Thompson, and their main emphasis
53
was to look at centralizers and normalizers of prime power order subgroups
and analyze their relationships. A new term was coined for the normalizer of a
nonidentity prime power subgroup, and that was local or plocal subgroup (p
being the prime power). Thus, the techniques of local analysis are the methods
of examining local subgroups.
The local analytic methods were explored further by Thompson in his
classification of minimal simple groups, and his Agroup theorem of 1968. An
Agroup is a simple group whose local subgroups are each solvable. Thompson
explored all possible simple groups fitting this description, and was able to
classify the Agroups:
Theorem 4.1 If G is a simple Ngroup, then G is isomorphic to one of the
following groups:
PSL2(q), where q > 3,
Sz{q), where q 22n+1, n > 1,
PSL3(3), U3(3), 2F4(2)', A7, or Mu.
U3(S) is a unitary group, and 2F4(2)' is a Ree group of Lie type. Thompsons
strategy was to show that an arbitrary Agroup has internal structure that
looks like one of the groups listed. Then resemblance was shown to be actual
isomorphism. This process is mirrored in the classification theorem, as will be
seen below. One concept that was invaluable to Thompsons Agroup theorem
and later to local analysis in general was the idea of embedded subgroups. An
example of a type of embedded subgroup is a strongly embedded subgroup
M of G This means that \M\ is even and the following hold:
54
1. Ca(t) CM for every involution t of M
2. Nq(S) C M for each Sylow 2subgroup S of M.
Strongly embedded subgroups themselves were actually classified by Bender
in 1971 as either PSL2(q), U^{q), or Sz(q) for q even.
While local analysis was developed, and results of a different na
ture were obtained because of the change of emphasis, there were also further
changes in direction by creative individuals. A couple of these different ap
proaches are mentioned now. The method of both the FeitThompson theorem
and many classification theorems that followed was generally to look at min
imal counterexamples and either derive a contradiction to the theorem state
ment, or show isomorphism of the group in question to a known simple group.
The procedure to achieve this was to examine relatively small subgroups to
develop the local subgroup structure. Helmut Bender changed this approach
in his attempt to simplify the proof of the FeitThompson theorem. He stud
ied the intersections of maximal subgroups which contained the centralizer of
some involution. This approach is called the Bender method, and was used to
dramatically reduce the complexity of such theorems as Walters result about
abelian Sylow 2subgroups, and Gorensteins and Walters result about dihe
dral Sylow 2subgroups. Originally, Bender was looking for a revision of the
classification as a whole, beginning with the FeitThompson theorem, but his
method became a useful tool in itself.
Another innovation that was second only to local analysis techniques
55
was Fischers internal geometric analysis. We have seen the work of Fis
cher with respect to the discovery of sporadic simple groups. His ideas of
3transpositions went much further than only the discovery of his sporadic
groups however. Recall that a class of 3transpositions is a conjugacy class of
involutions where the product of any two has order 1, 2, or 3. Also, the group G
in question is generated by this conjugacy class. Fischers geometric approach
was to consider a graph whose vertices are the elements of the conjugacy class,
and any two elements which commute with each other are connected by an
edge. The group G acts as a group of automorphisms of the graph since under
conjugation, G permutes the vertices of the graph but preserves the incidence
relation on the graph. Thus, Fischer saw that the structure of the group G
is related to the geometry of the graph. His work inspired others such as As
chbacher, and the definitions of connected and nonconnected came from the
nature of the graphs. We will see that these play a very important role in the
classification theorem.
4.2 The Theorem
The entire classification theorem is a monumental enterprise of be
tween 10,000 and 15,000 pages, taken from around 100 contributors, and writ
ten over a period of more than 30 years. There are articles stretching out
among perhaps 500 journals that comprise the theorem. The main contrib
utors are an international group mainly from the U.S., Germany, England,
Canada, Australia, and Japan. Results were collected starting around the
late 1940s and complete classification was obtained in the early 1980s. We
56
have seen that a systematic approach to the classification was proposed as
late as 1972. The theorem itself states that all finite simple groups have been
found. That is, any finite simple group is isomorphic to one of those already
discovered:
Theorem 4.2 Main Classification Theorem: Every (nontrivial) finite simple
group is isomorphic to one of the following:
1. A group of Lie type
2. An alternating group
3. One of the above mentioned 26 sporadic groups (see p42).
The general structure of the theorem is that of induction. A minimal simple
counterexample G is chosen such that G is assumed not isomorphic to any
known simple group, and any simple group with order less than G is a known
simple group. Also, suppose that the group G has a set of properties, X.
Given this information, one can prove that G is actually a known simple group,
deriving a contradiction. The inductive nature of the proof is important for
looking at internal properties of subgroups of G. For example, there is a result
which states that:
Theorem 4.3 Given a minimal simple counterexample G with a set of prop
erties X, if H C G and K < H, then H/K is a simple group with properties
X.
An alternate form of the classification theorem which makes the inductive
nature obvious is the following:
Theorem 4.4 Main Classification Theorem, alternate form: If G is a finite
simple group each of whose proper subgroups is a known simple group, then G
57
is a known simple group.
Suppose that we have a minimal counterexample G with X properties
that we assume is not a known simple group. This assumption forces us
to consider the internal properties of G to be as complicated as any finite
group. We cannot assume that G looks like a known simple group from the
start, for that is what we are trying to prove. The next step is to force
our counterexample to look like a known simple group. Obviously, this is
not an easily accomplished task and many of the high powered local analysis
techniques must be used carefully in the examination of the internal structure
of the group. There are many possibilities for a group G with X properties,
and each must be considered. This accounts for much of the complexity and
length of the theorem, since there are around 100 different paths to follow
to show that G looks like a known simple group. The paths themselves are
determined by the properties of G, so each case is different. The classification
theorem is complete in that it exhausts all of the possible structures of G and
leads all possible simple groups to the structure of a known simple group. In
order for us to know that our simple group looks like a known simple group,
we must have a very detailed description of the known simple groups. This
part of the theorem is called the recognition theorems. Once it is determined
that G looks like a known simple group, then the steps toward isomorphism
must be taken. That is, internal resemblance must be shown to be actual
isomorphism.
It is to be noted that the structure of the classification theorem is
very similar to that of the FeitThompson theorem. In fact, one can break
58
down the process of both into three steps: [10]
1. ) Use the given properties of G to determine the structure and
embedding of maximal subgroups containing or intersecting centralizers of
involutions by local analysis.
2. ) Eliminate as many of these possible configurations by using char
acter theory on smaller groups, local analysis on larger groups, and arithmetic
methods.
3. ) Use recognition theorems (generators and relations) to prove that
the only possible configuration left is isomorphic to a known simple group.
Beginning with the last step first, each of the known simple groups
must be recognizable by some defining feature. These recognition theorems
usually are in terms of generators and relations, especially for the groups of
Lie type. The alternating groups can also be characterized by generators and
relations, as the following theorem shows:
Theorem 4.5 If the group G is generated by the elements x\, x2,..., xn_2
subject only to the relations xf = 1, x\ = 1 for 2 < i < n 2, (rEjZj+i)3 = 1 for
1 < i < n 3, and (XiXj)2 = 1 for 1 < i < n 4 and i + 1 < j, then G = An.
The recognition theorems for the sporadic groups usually depend on
how the sporadic group was constructed. For example, those sporadic groups
which were constructed by their centralizer of an involution can be character
ized by this centralizer. (Theorems 3.4 and 3.5 are examples of recognition
theorems). Those sporadic groups which are rank 3 are characterized by their
one point stabilizers. (See p.49) Thus, much of the discussion in Chapter two
serves to describe some of the recognition theorems. If the counterexample
59
group G* is shown to have such characteristics as are given in the recognition
theorem of group G, then the purpose of the recognitions theorems is to state
that G* is necessarily isomorphic to G.
The first two steps of the classification theorem are then to prove that
G* has some defining features that are in one recognition theorem. We have
seen that centralizers of involutions and Sylow 2subgroups play an important
role in the internal structure of any simple group. Many sophisticated features
of a group have been discovered in relation to these two. One of the reasons for
this is that Sylow 2subgroup structure depends on the properties of centraliz
ers of involutions (since Sylow 2subgroups contain all of a groups involutions,
and there is always an involution in the center of a Sylow 2subgroup), and
centralizers of involutions can often lead to recognition theorems. There are
complicated techniques to achieve this leap however, including what are called
fusion arguments. The purpose of this line of theory is to give precise descrip
tions of the way in which involutions in a Sylow 2subgroup are conjugate in the
group. Some of the famous results are Glaubermans Z* theorem, Thompsons
fusion lemma, and Alperins fusion theorem. Embedding is another property
of subgroups which developed into important theory. What are called sig
nalizer functor methods grew out of the study of embedded subgroups. The
accumulation of all of the possible internal structures of a simple group can be
summarized in the four part division of the main classification theorem proof:
The classification of nonconnected simple groups,
The classification of connected simple groups of component type,
The classification of small simple groups of characteristic 2 type
60
The classification of large simple groups of characteristic 2 type [10].
The definitions of each of these are quite involved. We have seen how
connected and nonconnected groups might arise. Let us now define character
istic 2 type.
Definition A is characteristic 2 type if F*(H) is a 2group for every 2local
subgroup H of X.
Now F*(H) is called the generalized fitting subgroup of X and
F\H) = L(X)F(X).
F(X) is the fitting subgroup of X, which means it is the unique largest
nilpotent subgroup of X.
L(X) is the layer of X, which means that L(A) is the product of
all subnormal quasisimple subgroups of X, with L(X) = 1 if no subnormal
subgroups exist.
A subnormal subgroup of A is a subgroup Y such that Y Y\ <3
Y2 < ... < Yn X for appropriate subgroups Yi of A.
A quasisimple subgroup of A is a subgroup K such that K = [K, K]
and K/Z(K) is simple.
[K, K] = ([A:, k]\ke K) where [k, k] = k^k^kk.
This gives a glimpse of the complexity involved in pinning down the
internal structure of a simple group. The four part division above can actually
be reduced to two parts, that concerning noncharacteristic 2 type, and that
concerning characteristic 2 type groups.
We have taken a rather nontechnical look at the enormous theorem
as Gorenstein refers to it. Hopefully this will serve as at least an introduction
61
to the main objective and some methods of the proof. A revision of the proof
has been suggested, and begun. It was spearheaded by Daniel Gorenstein
who unfortunately died in 1992. With such a large proof to begin with, it
is generally held that completely new techniques would have to be obtained
before any remarkable reduction in length could be realized. When the theorem
was nearing completion, a headline in the New York Times read, A School
of Theorists Works Itself Out of a Job, 1980. Yet all of those involved in
the proof had positive ideas of the future of group theory. Gorenstein cited
applications to such fields as mathematical logic and number theory due to
the classification theorem [11]. The relationship between finite group theory
and finite geometries was mentioned by Aschbacher as possibly benefitting
from the classification theorem [1]. Also even within the field of group theory,
many felt there was much to do. As Gorenstein comments, the obituary for
finite group theory has been totally premature [9]. The theorem itself is a
testament to the perseverance and cooperative nature of human kind. It has
been said in reference to the length and complexity of the theorem that either
they have been a bit dim in finding the most effective techniques to prove the
classification theorem, or they have been very clever indeed.
62
REFERENCES
[1] M. Aschbacher, The Finite Simple Groups and Their Classifi
cation, Yale University Press, 1980.
[2] , Finite Group Theory, Cambridge University Press, 1986.
[3] R. W. Carter, Simple Groups of Lie Type, John Wiley and Sons,
1989.
[4] M. J. Collins, Finite Simple Groups II, Academic Press Inc., 1980.
[5] J. Conway, R. Curtis, S. Norton, R. Parker, and R. Wilson,
Atlas of Finite Groups, Clarendon Press Oxford, 1985.
[6] J. Gallian, The Search for Finite Simple Groups, Mathematics
Magazine, 49 (1976), pp. 163179.
[7] , Contemporary Abstract Algebra, DC Heath and Co., 1994.
[8] D. Gorenstein, Finite Simple Groups and Their Classification,
Israel Journal of Mathematics, 19 (1974), pp. 566.
[9] , Finite Simple Groups, An Introduction to their Classifica
tion, Plenum Press, NY, 1982.
[10] , Classification of Finite Simple Groups Vol I, Plenum Press,
NY, 1983.
[11] , The Enormous Theorem, Scientific American, 253 (1985),
pp. 104115.
[12] M. Hall Jr., A Search for Simple Groups of Order Less than
One Million, in Computational Problems in Abstract Algebra, J. Leech,
ed., Pergamon Press, NY, 1969, pp. 137168.
[13] I. M. Isaacs, Algebra A Graduate Course, Brooks/Cole Publishing
Co, 1994.
63
[14] M. B. Powell and G. Higman, Finite Simple Groups, Academic
Press Inc., 1971.
[15] R. Silvestri, Simple Groups of Finite Order in the Nineteenth
Century, Archive for the History of Exact Sciences, 20 (1979), pp. 313
356.
[16] I. Stewart, Galois Theory, Chapman and Hall, 1989.
[17] H. Weyl, The Classical Groups, Princeton University Press, 1946.
64

PAGE 1
A HISTORY OF FINITE SIMPLE GROUPS by Faun C.C. Doherty B.A., Oberlin College, OH, 1993 A thesis submitted to the University of Colorado at Denver in partial fulfillment of the requirements for the degree of Master of Science Applied Mathematics 1997
PAGE 2
This thesis for the Master of Science degree by Faun C.C. Doherty has been approved by J. Richard Lundgren William E. Cherowitzo Stanley E. Payne Date ____________________ __
PAGE 3
Doherty, Faun C.C. (M.S., Applied Mathematics) A History of Finite Simple Groups Thesis directed by Associate Professor J. Richard Lundgren ABSTRACT A group is a set together with an associative binary operation such that there exists an identity element for the set, and an inverse for each element in the set. All finite groups can be broken down into a series of finite "simple groups" which have been called the building blocks of finite groups. The history of finite simple groups originates in the 1830's with Evariste Galois and the solution of fifth degree polynomial equations. In the twentieth century, the recognition of the importance of finite simple groups inspired a huge effort to find all finite simple groups. This classification project was completed in 1981. We shall begin by taking a historical look at the earliest methods of analyzing the structure of finite groups according to their order. Finite simple groups can be divided into two types, those belonging to infinite families and the 26 sporadic simple groups. We shall look at the discovery and representation of many of these. Finally, we shall discuss the monumentallO,OOO to 15,000 page proof of the classification of all finite simple groups. This abstract accurately represents the content of the candidate's thesis. I recommend its publication. SignedJ. Richard Lundgren lll
PAGE 4
ACKNOWLEDGEMENTS I would like to thank Professor Lundgren for his support in writing this thesis. Also, thanks to my parents for their example and Michael for his patience.
PAGE 5
CONTENTS Chapter 1 Introduction 2 The Range Problem 3 2.1 Introduction to the Problem 2.1.1 Sylow's Theorems ..... 2.1.2 Other Theorems, Corollaries, Etc. That Will Prove Useful: ... 2.2 Some History 2.2.1 Holder ... 2.2.2 Cole, Burnside 2.2.3 The Completion of the Range Problem Through Order One Million 2.3 Some Examples The Simple Groups 3.1 Infinite Families of Simple Groups 3.1.1 The Alternating Groups ..... 3.1.2 Simple Groups of Lie Type .. 3.1.2.1 The Classical Linear Groups 3.1.2.2 Other Lie Groups .... 3.2 The Sporadic Simple Groups v 1 3 3 4 5 7 7 12 16 19 30 30 30 35 36 39 40
PAGE 6
3.2.1 The Mathieu Groups ....... 42 3.2.2 Centralizer of Involution Problems 43 3.2.3 Rank 3 Permutation Groups ... 48 3.2.4 The Remaining Sporadic Simple Groups 49 4 The Classification Theorem 51 4.1 History 0 51 4.2 The Theorem 56 References 0 0 0 0 63 Vl
PAGE 7
1. Introduction Some have referred to the study of simple groups as the 'ElDorado' of finite group theory. It has been a very active field of study through the twentieth century and has its roots in the nineteenth, as does group theory itself. A group is defined as a set together with an associative binary operation defined such that there exist an identity element for the set, and inverses for each element of the set. The set is closed under the operation. A normal subgroup H of a group G is a subgroup such that aH = H a for all a E G. Another definition of normal is that a1 H a= H for a E G. A simple group is a group which has no normal subgroups except itself and the identity (which are always normal). Those groups with prime order have no subgroups except for the identity and the group itself, thus they are considered trivially simple. For the rest of this paper, the term simple group will refer to finite nontrivial simple groups. Simple groups are special kinds of groups that are the building blocks of all other groups, thus the importance in their study. This idea was recognized as early as 1832 by Evariste Galois, and later a search for the simple groups took place. In the twentieth century this search culminated in a monumental theorem which classifies all simple groups. One of the earliest methods of locating simple groups is called the range problem. This is a systematic examination of the internal structure of groups according to the order of the group. Chapter one of this paper will outline the history of this 1
PAGE 8
problem and the methods used through the analysis of groups up to order one million. The second chapter will describe the simple groups by their types: infinite families of simple groups and the sporadic simple groups. How some of these groups can be represented as well as the methods of their discovery will be discussed. Finally, a general outline of the classification theorem will be given in the last chapter. 2
PAGE 9
2. The Range Problem 2.1 Introduction to the Problem Among the methods of determining all finite simple groups, the ap proach of examining individual groups of certain orders can seem at times slow and methodical. Yet this task, begun in 1892 by Otto Holder, has proven fruit ful in the advancement of group theory, if not always in the discovery of new simple groups. It has shed a great deal of light upon the structure of groups with given orders which allows one to understand the nature of simple groups, at least in so far as determining what they are not. This particular prob lem lasted through to 1975 when Marshall Hall, Jr. completed the individual examination of groups with particular orders through the order of 1,000,000. About eleven individuals from 1892 to 1975 participated in the solution of this problem, each aided by the work and discoveries of those who came before. The range problem itself is not difficult to understand, in light of the search for simple groups. It is simply this: given a particular natural number, say n, what can we say about the structure of any group having n elements? And in particular, can we determine if the group has any normal subgroups besides itself and the identity, i.e., can we show that the group is not simple? If the group is simple, is it unique? Through the history of this problem, there were two main methods used to explore the structure of groups with a given order. One was to use the Sylow theorems and the other was to employ 3
PAGE 10
character theory. It will be the task of this paper to concentrate only on the Sylow theorem methods, thus a word about these theorems is in order. 2.1.1 Sylow's Theorems Ludvig Sylow, a Norwegian mathematician came up with the Sylow theorems in 1872 by way of the study of permutation group theory. These results lost no importance with the development of abstract group theory, in fact, their importance grew. The Sylow theorems as we state and prove them today are based on the fundamental concept known as Lagrange's theorem, and it is here that we shall start. Theorem 2.1 [Lagrange's theorem] Suppose H C G zs a subgroup. Then IGI = IHIIG: HI. Note that I G : HI is the index of H in G, or the number of distinct right cosets of H in G. A right coset is the set Hg = {hg I h E H} where H G. We can show easily that the group G is the disjoint union of the distinct right cosets. The cardinality of each coset is equal to the number of elements in the subgroup H, and with these two facts, we may deduce that the order of G is the order of H times the number of distinct right cosets that partition G. Theorem 2.2 [Sylow's Theorem 1] If pk I IGI, then G has at least one subgroup of order pk for any prime p. Thus if any power of a prime divides the order of our group, then the group has a subgroup of order that power of the prime. Theorem 2.3 [Sylow's Theorem 2] If H G, and IHI = pk then H zs 4
PAGE 11
contained in some "Sylow psubgroup ". A "Sylow psubgroup" is a subgroup of G such that its order is equal to the full power of p in the order of G. For example, if we have a group of order 24 5 112 a sylow 2subgroup would have order 24 The set of all Sylow psubgroups of G is denoted Sylp(G). We know from the first Sylow theorem that Sylp( G) is not empty. We can also find a Corollary ( (2.8) below) which states that if only one Sylow psubgroup exists, then it is normal in G. This fact will allow us to eliminate easily many integers as possible orders of simple groups. Theorem 2.4 [Sylow's Theorem 3] The number of Sylow psubgroups of G, z.e., ISylp(G)I(written np) has the following properties: np 1 mod p and np 1 mod pe if pe:::::; IS: S n Tl for all S and T E Sylp(G) with S 1T. The examination of the structure of groups with a given order is feasible be cause of a number of other results besides the Sylow theorems, although many of these results are based on the Sylow theorems. A number of these results shall be listed below and referred to throughout this chapter. 2.1.2 Other Theorems, Corollaries, Etc. That Will Prove Useful: Theorem 2.5 A nontrivial finite pgroup has a nontrivial center. A pgroup (where p is prime) is defined as a group in which every element has order a power of p. The center of a group (Z(G)) is a normal subgroup of G composed of all elements which commute with all other elements of G. 5
PAGE 12
Theorem 2.6 If IGI = pa, where pis prime and a> 1, then G is not simple. Proof: Let IGI = pa, and suppose that G is simple. Since G itself is apgroup, by (2.5) we know that 1 < Z(G) 0, m > 1 and p does not divide m. If G is simple, then np (G) satisfies all of the following: 1. np divides m 2. np 1 mod p 3. IGI divides (np!) Corollary 2.10 Let P be a Sylow psubgroup of G. Then np = IG: Na(P) I, and np divides IG: Pl. Theorem 2.11 Let H G with IG: HI = n. Then there exists N 1 and IGI does not divide n!, G is not simple. Corollary 2.12 Let H G and IG: HI = p where p is the smallest prime divisor of IGI. Then H
PAGE 13
there are exactly cp( n) different isomorphisms that map B to C. Theorem 2.14 Every two Sylow psubgroups of G are conjugate. 2.2 Some History 2.2.1 HOlder The range problem was initiated by Otto Holder (18591937) in 1892. Before 1892, Holder published two papers that considerably contributed to the emphasis on this problem. The first was published in 1889 in the Mathema tische Annalen [15]. It was a paper primarily dealing with the solution of equations. However, what was evolving into group theory, thanks to Evariste Galois, who we will discuss in chapter 2, seems to have proved useful to his work. The concepts of normal subgroups and a composition series are dis cussed. A composition series is a series of normal subgroups 1 = Go
PAGE 14
a finite number of operations [elements] could be known [15]." One other advancement of this time deserves recognition. Group theory was evolving into a subject in its own right, and the idea of treating groups in the abstract, an idea attributed to Cayley, was finally being accepted. In his start upon the range problem, Holder was the first to study groups in the abstract. More often in the past, groups were considered with respect to their mode of representation, for example, a linear transformation group. The range problem initiated the type of exploration that only required knowledge of the order of the group. Holder studied groups having orders from 1 to 200. He did not dis cover any new simple groups, since the unique simple group of order 60 was known to be simple (it is A5 and will be discussed below), as was the group of order 168 (PSL2(7), found by Jordan in 1870). His methods were important, however, since they were used by all others working later on the range prob lem. His ideas provided important general theorems which can be, and were used within any range, and will be discussed below. The most useful tools that Holder employed were the Sylow theorems. Holder was comfortable with permutation groups, and also used this theory. Many of the lemmas that he used in more general theorems came from permutation group theory combined with the results of Sylow's theorems. One of his general theorems has to do with groups that have orders equal to a product of three or fewer primes, not necessarily distinct. Holder proved that groups with orders pq, p2q, or pqr are not simple. Sylow had already taken care of those groups with orders pa (2.6). These theorems can be proven in a more effective 8
PAGE 15
manner using only the Sylow theorems, which Burnside did in later years. The following proofs are similar to the methods used by Burnside, rather than the permutation theory used by Holder. HOlder's Proofs Using the Sylow Theorems Theorem 2.15 If IGI = pq, where p and q are primes, then G is not simple. Proof: Let IGI = pq, where p and q are primes, and assume G is simple. Without loss of generality, we may assume that p > q. Then the only choice for np is np = q, since np must divide q by (2.9), but cannot equal 1 by (2.8) and our assumption. This implies that q 1 mod p, which is a contradiction since p > q. Thus, our assumption is false, and G is not simple. Theorem 2.16 If IGI = p2q, where p and q are primes, then G is not simple. Proof: Let IGI = p2q, where p and q are primes, and assume G is simple. The choices for nq are: nq = p or p2 Suppose that nq = p. Then p 1 mod q so p > q. But the only choice for np is q which implies q 1 mod p, thus, q > p. So nq #p, which means that nq = p2 Let us now count elements in the group. Since nq = p2 we have p2 subgroups each with order q. Notice that they have prime order, which means that they are cyclic, and have no two elements in common except for the identity. This means that there are p2(q1) elements with order q. Let 8 denote the number of the rest of the elements. Then Thus, there are enough elements not of order q to only fit into one Sylow psubgroup, which means there is a unique Sylow psubgroup, which must be normal in G. But this is a contradiction to our assumption that G is simple, which leaves us only with the alternative that one of the Sylow subgroups is 9
PAGE 16
umque, thus normal (2.8). Thus, our assumption was wrong, and G is not simple. Theorem 2.17 If IGI = pqr, where p q and r are przmes, then G is not simple. Proof: Assume IGI = pqr, where p, q and r are primes, and assume G is simple. Without loss of generality, we may assume that p > q > r. The possibilities for the size of Sylp (G) are as follows: np = q,r, or qr nq = p, r, or pr nr =p,q, or pq. Notice that we may eliminate q and r as possibilities for np since p > q > r (using Sylow's 3rd (2.4)). Also, we may eliminate r as possibility for nq for the same reasons. We may eliminate q as possibility for nr since we know that IGI cannot divide q! since there is no p factor in q!. We conclude that there are four cases only: 1. nq =p 2. nq = p 3. nq = pr 4. nq =pr nr =p nr =pq nr =p nr =pq. If we examine each of these cases by counting elements, we find that none are feasible. The First Case: we can conclude that the number of elements with order p is qr(p1), the number of elements with order q is p(q 1) and the number of elements with order r is p(r 1). Note that this is possible since each Sylow subgroup has prime order, so no two Sylow subgroups of the 10
PAGE 17
same order have elements in common except for the identity. If we add the number of elements that we have so far, it is qr(p1) + p(q1)+ p(r1) = pqrqr + pqp + prp or, pqrqr + pq + pr2p Note that qr + pq is positive since p > r, and pr2p is positive if r > 2 (zero otherwise). Thus, we have pqr+ some positive number as the number of elements in G, which is a contradiction. Thus, the number of Sylow subgroups is not the first case. The Second Case: Using the same arguments as above, the second case provides us with the following number of elements: qr(p 1) + p(q 1) + pq( r 1) but this is equal to pqr qr + pq p + pqr pq = pqr + pqr qr p or, pqr + qr(p1)p and we see that qr(p1)p must be positive. Thus, again we have over pqr number of elements, which is a contradiction. The Third Case:Vsing the same arguments as above, the third case provides us the following number of elements: qr(p1) + pr(q1) + p(r1) which is equal to pqr qr + pqr pr + pr p = pqr + pqr qr p and this is identical with the second case, and thus a contradiction. The Fourth Case: The same counting technique provides us with the following number of elements: qr(p1) + pr(q1) + pq(r1) which equals pqr qr + pqr pr + pqr pq = pqr + pqr qr + pqr p( q + r) and note that pqr qr is positive, as is pqr p( q + r) if q + r < qr, which is true if q and r are 2: 3 and 2 respectively, which they are. Thus we have another contradiction, which implies that our original hypothesis was incorrect, and G is not a simple group. The power of these theorems, along with a few others, eliminated all but seven orders out of the first 200 cases. The seven remaining groups had 11
PAGE 18
orders 60 (known to be simple), 90, 112, 120, 144, 168 (known to be simple), and 180. Holder was able to show that all but 60 and 168 were orders of non simple groups using various techniques of permutation group theory, yet it has been said that his ability to use permutation groups was somewhat lacking. It did take him nearly twenty pages of calculation to demonstrate that groups of order 144 and 180 were not simple. 2.2.2 Cole, Burnside It was an American mathematician who followed the path laid by Holder. Frank Nelson Cole (18611927) continued the range problem in 189293 examining groups with orders ranging from 201 to 660. The methods used by Holder were also used by Cole. The Sylow theorems provided the most powerful tool of investigation, and Cole also looked at groups in the abstract sense, "only recurring when convenient to their representation in terms of sub stitutions of n letters [permutation groups] [15]." Holder's theorems of three or fewer primes proved useful to eliminate all but 84 groups between 201 and 500. Sylow's theorem that np 1 mod p eliminated another 56. Eventually, Cole determined that A6 PSL2(11), and PS2(23), groups of orders 360, 660 and 504 respectively, were the only simple groups with order between 201 and 660. The simple group of order 504 was never recognized as simple before Cole's work, even though it had been discussed by mathematicians such as Mathieu and Kirkman. It was classified later as PS2(23 ) following the ad vancements made by Dickson and Moore. It was a special discovery in more 12
PAGE 19
ways than one, since it launched the work of Eliakim Hastings Moore (18621932) who discovered that the infinite family of groups, PSL2(pn) was simple except when pn = 2 or 3. This in turn led to the proof by Dickson that the infinite family of groups PSLm(Pn) are simple, which is a generalization of Jordan's original 1870 result. This family shall be discussed further in the subsequent chapter. Notice that there were no new methods evident in Cole's work, the Sylow theorems served him well. William Burnside (18521927), who has been called the first real group theorist in history because of his dedication to abstraction, was the next mathematician to work on the range problem. Once again, his techniques did not stray far from the Sylow theorems and permutation group theory. He did develop some arithmetic tests, the most important of which states that a simple group of even order must be divisible by either 12, 16, or 56. The understanding of permutation groups had advanced since Holder's and Cole's work, which was a help in Burnside's pursuits. Ironically though, Burnside was very active in rewriting theorems previously based on permutation theory using only the abstract ideas such as conjugacy classes and normalizers. Burn side claimed that even in reference to the proofs of the Sylow theorems, "from the point of view of the right method they leave something to be desired [15]." He subsequently rewrote them. Notice that the proofs given above of Holder's three or fewer primes theorem are essentially Burnside's rewrites. Not only did Burnside simplify the proofs for these, but he also extended the theorem to include combinations of four or fewer primes. A couple of his proofs are given below. 13
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Burnside Theorems of Four or Fewer Primes Theorem 2.18 If IGI = p3q, where p and q are primes, then G is not simple. Proof: Assume IGI = p3q, where p and q are primes, and assume G is simple. The choices for nq are: nq = p p2 or p3 and np = q, which implies that q > p. Suppose that nq = p. Then p 1 mod q which contradicts q > p. Suppose that nq = p3 Count elements: there are p3 subgroups, each with order q, which have trivial intersections. Thus there are p3(q1) elements with order q. Let 8 denote the number of the rest of the elements. Then I G I = e + p3 ( q 1)' or e = p3 q p3 ( q 1) = p3 Thus, there are enough elements not of order q to only fit into one Sylow psubgroup, which means there is a unique Sylow psubgroup, which must be normal in G. But this is a contradiction to our assumption that G is simple, which leaves us with the last possibility. Suppose that nq = p2 Then p2 1 mod q ::::} q I (p2 1) ::::} q I (p + 1) (p1). Since q is prime, this implies that q I (p + 1) or q I (p1). Since q > p q I (p + 1) only. But this implies that p < q p + 1 so q = p + 1 and p and q are consecutive primes. But the only consecutive primes are 2 and 3, so if G is indeed simple, p = 2 and q = 3 is the only possibility. Thus, if we show that a group of order 23 3 is not simple, we have a contradiction. Suppose IGI = 23 3. Then n2 = 1 or 3. But note that IGI does not divide 3!. Thus n2 = 1, which implies that G is not simple. So the only possibilities left are that one of the original Sylow subgroups is unique, thus normal, contradicting our hypothesis that G is simple. Theorem 2.19 If IGI = p2q2 where p and q are przmes, then G is not 14
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simple. Proof: Assume IGI = p2q2 where p and q are primes, and assume G is simple. Without loss of generality, we may assume that q > p. The choices for nq are: nq = p and p2 If nq = p, then p 1 mod q, which contradicts the fact that q > p. So suppose that nq = p2 Then the argument from the above proof holds, i.e., p2 1 mod q::::? q I (p2 1) ::::? q I (p + 1)(p1). Since q is prime, this implies that q I (p + 1) or q I (p1). Since q > p q I (p + 1) only. But this implies that p < q p + 1, so q = p + 1 and we can show that a group of order 22 32 is not simple: Let IGI = 22 32 Then n3 = 1, 2, or 4. But 2 is not 1 mod 3, and IGI does not divide 4!, thus the Sylow 3subgroup is unique, thus normal in G. So the only possibilities left are that one of the original Sylow subgroups is unique, thus normal, contradicting our hypothesis that G is simple. In 1895, Burnside completed the range problem up to order 1092. Shortly after this time, beginning in 1896, a new technique emerged developed by Burnside and Georg Frobenius (18491917) called character theory. This theory, which is based on the study of certain functions ("characters") from a group into the complex numbers, has made a great impact on the study of simple groups through this century. It was character theory that provided Burnside with a proof of a monumental theorem that follows and outshines the four or fewer primes result. In 1904, Burnside proved that any group with order paqb where p and q are prime is not simple, unless it is of prime order [6]. Obviously, this theorem plays a significant role in the simplification of the 15
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work required on the range problem after 1904. The extension of this result to orders made up of a combination of three primes, paqbrc, has been a difficult problem which has lasted until the present, and the method of investigation has most often been character theory. Unfortunately, it is beyond the scope of this paper to mention character theory in more depth. 2.2.3 The Completion of the Range Problem Through Order One Million The turn of the century saw two mathematicians, George Abram Miller (18631951) and his student G. H. Ling work on the range problem for orders between 1092 and 2001 in 1900. The original techniques of investigation had not changed much, however there were a couple of new results which came from the older methods. One was that any group of order paq, paq 2 and paqb (for a = 1, ... 5; p < q) was not simple. Notice that these results were the previews of what was to come in Burnside's 1904 theorem. The problem of odd versus even orders was well under investigation at this time, as we shall examine in the next chapter. The result at this time which was put to good use was the fact that there were no simple groups with odd orders less than 2835. There was increased work on the theory of permutation groups, and on transitive groups in particular which helped with the investigation of individual orders. A permutation group on a set is called transitive if for each pair of elements of the set, there exists an element in G which sends one to the other. With these techniques, Miller and Ling showed that there was no simple group between 1093 and 2000. There seemed to be quite a gap after the work of Miller 16
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in interest in the range problem. It was not until1912 that anyone approached the orders following 2000. This may have been due to the difficulty that the larger orders presented, and the lack of new results which would act quickly and sweepingly, although one must remember the Burnside theorem which did exactly that. It was not until 1954 that new methods actually arose to handle groups of particular orders. While work on the infinite families of simple groups was taking place, there was a bit of a lull in the advances on the range problem during the early twentieth century. In fact, work on the range problem was sporadic through the twentieth century. L. P. Siceloff was the next mathematician to tackle the orders 2001 through 3640 in 1912. He found simple groups with orders 2448, 2520 and 3420. He was not able to prove the uniqueness of the simple group with order 2520, and it was not until 1922 that Miller successfully showed that the group was A7 and unique. Cole came back to the game in 1924 with the orders 3641 through 6232. He found four simple groups having orders 3420, 4080, 5616, and 6048. He found difficulty with the uniqueness of two orders, 5616 and 6048. Both of these are unique simple groups, as shown by Richard Brauer in 1942 using character theory. It took eighteen years to find the methods to complete this task! The next time that someone chipped away at the range problem was in 1963. Michaels took the task of showing that the unique simple groups between 6233 and 20,000 were of orders 7800, 7920, 9828, 12,180, and 14,880. In 1972, Marshall Hall, Jr (19101990) extended the range problem to order one million [12]. He drew together all of the methods used from the 17
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late nineteenth century onwards, a great deal of the later methods relying on advanced techniques of character theory. His assortment of methods also included some computer work. Hall's methods were unsuccessful with only 21 orders. It was in 1975 that two students, Beisiegel and Stingl, extended work on the classification of simple groups according to the size of their Sylow 2subgroups undertaken by Paul Fong. The remaining 21 orders were taken care of, and the range problem to one million was complete. It was not necessarily the people working on the range problem that discovered new simple groups. In fact, not many new simple groups were found at all during the course of the range problem. In 1900, Dickson listed a total of 53 known simple groups, many members of infinite families of simple groups (see below). By 1972, only three new groups were added to this list. M. Suzuki discovered the simple group with order 29,120 in 1960 as he discovered the infinite family, Sz(2n). Z. Janko uncovered the simple group of order 175,560 in 1966, however this group was not a member of an infinite family (that is, it is a sporadic simple group). In 1967, Hall and Janko discovered a simple group (J2 ) with order 604,800 which was also sporadic. None of these three groups was discovered because of work done on the range problem. Apart from these three, by 1900 those simple groups with orders less than one million were generally known to be simple before they were encountered in the course of the range problem. They consist of classical linear groups, alternating groups, and the Mathieu groups. 18
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2.3 Some Examples As examples of what the earlier work on the range problem was like, I have examined groups of various orders to demonstrate that they are not simple below. Easy violation of Sylow's third theorem (2.4), and use of Corollary (2.8): Example 1 If IGI =54, 587 = 132 17 19, then G is not simple: We only need to look at the possible number of Sylow 13subgroups to show that there is only one, thus it must be normal by (2.8). Note that by Lemma (2.9) the number of Sylow psubgroups must divide the remaining numbers left in the order of the group. Thus we have n13 = 1, 17, 19, or 17 19 (= 323). Only 1 1'mod 13, thus n13 = 1. Example 2 If IGI = 35,321 = 11 132 19, then G is not simple: This works in the same manner as above; we shall look for the number of Sylow 11subgroups to show that there can only be one: n11 = 1, 13, 132 19, 13 19, or 132 19. If we check each, none except 1 is 1 mod 11. If we had looked first at n13 we would have found that n13 could be 11 19 which is 1 mod 11. Example 3 If IGI = 7480 = 23 5 11 17, then G is not simple: This is an even order that works in the same manner. Notice the large number of possibilities for n17 : n17 = 1, 2, 4, 8, 5, 11, 10, 22, 20, 44, 40, 88, 110,220, or 440. 19
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However, none of these are 1 mod 17, thus the Sylow 17subgroup is solitary and normal. Easy violation of Lemma (2.9) Example 4 If IGI = 7260 = 22 3 5 112 then G is not simple: The possibilities for n11 are 1, 2, 4, 3, 5, 6, 12, 10, 20, 30, and 60. If we ignore 1 for the moment, we can exclude all possibilities except 12 by (2.4) So if we assume G is simple, then n11 = 12. But notice that IGI does not divide 12! since there is no second factor of 11 in 12!. Thus, by (2.9), we have a contradiction and G is not simple. Example 5 If IGI = 6468 = 22 3 72 11, then G is not simple: The possibilities for n11, excluding the smallest factors since they cannot be 1 mod 11, are: 1, 12, 14, 28, 21, 49, 147,98, 196,294, and 588. All except 1 and 12 violate (2.4). Thus, if we assume G is simple, then n11 = 12. Once again, IGI does not divide 12! since there is no second factor of 7 in 12!. Thus, by (2.9), we have a contradiction and G is not simple. Notice that for (2.9) to work, np must be fairly small. Here are a couple of examples where np is too large to use (2.9), and a different technique is needed: counting elements. Example 6 If IGI = 616 = 23 7 11, then G is not simple: Assume that G is simple. The possibilities for n11 are the following: n11 = 2, 4, 8, 7, 14, 28, 56. Notice that only 56 1 mod 11, so we may 20
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rule out the other possibilities. Can we also rule out 56 using (2.9)? No, 56 is large enough that IGI I 56!. Let us check n7 for an easier approach. n7 = 2, 4, 8, 11, 22, 44, 88. The only possibility that does not violate (2.4) is 22, and similarly, IGI I 22! since 22 is large enough. Thus, we have n11 = 56 and n7 = 22. A new strategy is needed for this problem. We know ISyl11(G)Iand ISyh(G)I and we know that each Sylow 11and Sylow ?subgroups have 11 and 7 elements in them respectively. Any group of prime order is also cyclic and we know that two different cyclic groups of the same order that have more than one element in common must be equal. Thus, each of the elements of Syl11 (G) and Syh( G) must intersect only trivially. We could count the elements in each. We have 56 groups with 11 1 distinct elements in each. The number of elements in Syl11 (G) is then 56(111), and similarly, the number of distinct elements in Syh(G) is 22(71). We have accounted for 56(111)+22(71) = 560+132 = 692 elements so far. There are only 616 elements in the group, so we have a contradiction. Thus, our assumption was incorrect, and G is not simple. Example 7 If IGI = 520 = 23 5 13, then G is not simple: This is similar to the above order. Assume that G is simple. Note the possibilities: n13 = 2, 4, 8, 5, 10, 20, 40 and n5 = 2, 4, 8, 13, 26, 104, 52. Us ing (2.4), we find that n13 = 40 and n5 = 26, and both numbers are too large to use (2.9). Noticing that the subgroups in Syh3(G) and Syl5(G) are of prime order, thus cyclic, we may count elements. We have 40(131) + 26(51) = 480 + 140 = 584 > 520. Thus, we have a 21
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contradiction, and G is not simple. The following two are more difficult cases using (2.9) and (2.4). Example 8 If IGI = 800 = 25 2 then G is not simple. Assume that G is simple. Notice that n5 = 2, 4, 8, 16,32 and only 16 1 mod 5. Also, IGI I 16!. So n5 = 16. Notice also that 16 is not 1 mod 52 so we may use (2.4), the later half, which states that there exists S and T E Syl5(G) such that S =!= T and 52 > IS: S n Tl by contrapositive. This implies that IS : s n Tl = 5. (This is because s n T is a subgroup of S and lSI = 52 thus if S n T =/= {1} which is necessary if 52> IS: S n Tl, then IS n Tl must be 5 or 52 It cannot be 52 because that would imply that S n T = S = T) By Lagrange's theorem, (2.1), we have that lSI= 52= IS: SnTIISnTI = 5 5 Thus, ISnTI = 5 and we may use Corollary (2.12) which states that since 5 is the smallest prime divisor of lSI, S n T
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implies that 24 divides INa(S n T)l. Thus, INa(S n T)l =52 24 or 52 25 If INa(S n T)l =52 25 then Na(S n T) = G, and S n T IS: S n Tl. Using the same process as above, we can conclude that IS: s n Tl = 3 and by Lagrange, IS n Tl = 32 By (12), since 3 is the smallest prime divisor of lSI, SnT
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33 25 since that would make G = Na(S n T) and thus not simple. Suppose INa(SnT)I = 33 24 Then IG: Na(SnT)I = 2 and since IGI does not divide 2!, G is not simple by (2.11). This is a contradiction to our assumption, thus INa(S n T)l 133 24 If n3(Na(S n T)) = 22 then INa(S n T)l = 33 22,33 23,33 24, or 33 25 We know that INa(S n T)l 133 24 or 33 25 Thus, suppose INa(S n T)l = 33 23 Then IG : Na(S n T) I = 22 and IGI does not divide 4!, showing that G cannot be simple (2.11). Suppose that INa(S n T)l = 33 22 Then IG : Na(S n T) I = 23 and still the index is too small, and IGI )' 8!. Thus since this is our last alternative, we conclude that our assumption was incorrect, and G is not simple. The following example uses a well known theorem, The "N/C Theo rem" (2.7). Example 10 If IGI = 792 = 23 32 11, then G is not simple. Assume that G is simple. The possibilities for n11 are the following: n11 = 2, 4, 8, 3, 9, 6, 12, 24, 18, 36, or 72. Only 12 1 mod 11, thus n11 = 12. Look at one subgroup in Syl11(G), sayS E Syl11(G). Let N be the normalizer in G of S, N = Na(S). Then since n11 = 12 = IG: Nl (2.10), we know that INI = 2 3 by Lagrange. Let C be the centralizer of Sin G, C = Ca(S). We know by the N/C theorem that the factor group N/C is isomorphic to a subgroup of Aut(S). The set of automorphisms of S has order cp(11) = 10, since Sis cyclic (2.13). This implies that IN: Cl divides 10. By Lagrange again, since INI = 2 3 = ICIIN: Cl, the only choice 24
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for IN: Cl is 2, thus ICI = 3 We see that the centralizer in G of S has Sylow 3subgroups. Let P E Syl3(C). Then IPI = 3. Consider Na(P) and note that Na(P) cannot equal G since we are assuming G is simple, and P
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Sylp (G) subgroups: n11 = 3,9,5,7,15,21,45,63,105,315 n5 = 3,9,7,11,21,33, 77,63,99,231,693 n7 = 3,9,5,11,15,45,33,99,55,165,495 n3 = 5, 7, 11, 35, 55, 77,385 The numbers in bold are those that do not violate either (2.4) or (2.9). These numbers indicate that the only possibilities for INc ( sp) I, where Sp E Sylp, by (2.10) are the following: INa(sn) I = 7 11 INa(ss) I = 3 5 11, or 3 5 1Na(s7) I = 3 7 11, or 5 7 INa(s3) I = 32 7, or 32 Working systematically, we shall try to eliminate each of these as possibilities. Suppose that 1Na(s5)l = 3 5 11. Look at ISylnl in Na(s5), denoted n11(Na(s5)) : n11(Na(s5)) = 1, 3, 5, or 15. Note that the only choice that does not violate (2.4) is n11(Na(s5)) = 1. Thus, by (2.10), 1 = INa(ss): NNa(s5)(sn(Na(ss))l and thus INa(ss)l = 3 5 11 = INNa(s5)(sn(Na(ss))l by Lagrange. But NNa(ss)(sn(Na(ss)) is the nor malizer in Na(s5 ) of a Sylow 11subgroup, and note that Na(s11 ) is the group of all elements in G that normalize a Sylow 11subgroup. Thus, NNa(ss)(sn(Na(ss)) Na(sn), which implies that 3 5 divides INa(sn)l. But we know that 1Na(s11) I = 7 11 from above, thus we have a contra diction. We now know that 1Na(s5 ) I = 3 5, and n5 = 231. Suppose that 1Na(s7)1 = 3 7 11. Note that n11(Na(s 7)) = 1,3, 7, or 21. By (2.4), 26
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n11(Na(s 7)) = 1 is the only possibility. Then by (2.10) and Lagrange, INNa(s1)(su(Na(s7))1 = 3. Butthisimpliesthat3IINa(su)l = 7, which is a contradiction. Thus, 1Na(s 7 ) I = 5 7 and n 7 = 99. Now look at the possibilities for n5(Na(s 7)): 1 or 7. By (4), n5(Na(s 7)) = 7, and by the same argument as above, this implies that 7 I Na(s5). We have from above that 1Na(s5)l = 3 5, thus we have a contradiction. The only possibility is that one of the Sylow subgroups is unique, thus normal. Therefore, G is not simple. The strategy of this last example is to use theorems about the size of Sylp( G) more than once to draw a contradiction. The following example starts in this manner, then requires a method previously seen, and comes to a conclusion with the same method used at first. Example 12 If IGI = 760 = 23 5 19, then G is not simple. Assume that G is simple. The following list the possibilities for the sizes of all Sylp(G): n2 = 5, 19,95 n5 = 2,4,8,38,76,152,19 n19 = 2,4,8,5,10,20,40 The numbers in bold indicate those that do not violate (2.4) or (2.9). The following are the possible orders of the normalizers of the Sylow subgroups 27
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by (2.10): 1Na(s2) I = 23 5, or 23 1Na(s5)1 = 2 5 INa(sig) I = 2 19 We would like to determine 1Na(s 2)1, so suppose 1Na(s 2)1 = 23 5. Then n5(Na(s2 )) = 1, 2, 4, or 8. We conclude by (2.4) that n5(Na(s2 )) = 1. Thus, using the same process as above, by (2.10) and Lagrange, we can conclude that INNa(s2)(s5(Na(s2))1 = 23 5. Since NNa(s2)(s5(Na(s2)) Na(s5 ), then 23 divides 1Na(s 5)l = 2, a contradiction. Thus, 1Na(s 2)1 = 23 and n 2 = 95. Note that 95 oj_ 1 mod 22 so by (2.4), there existS and T E Syl2 (G) such that S =J T and 22 > IS: S n Tl. This implies that IS: S n Tl = 2. By (2.10) we have that SnT
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The orders used for these examples are obviously fairly small. As one can guess, the larger the order, the more cumbersome are the choices for such numbers as np(G). Take the simple group, J1 for example. This group (described further below) has order 23 3 5 7 11 19. In order to determine n19 one must consider 56 possiblilities. Out of this 56, there are four numbers which cannot be eliminated using (2.4) or (2.9). Since 19 is the largest prime divisor of IGI, n19 should be the most accessible of all sizes of the Sylp(G) to find. Imagine what the others must be like! The shear magnitude of the problems increase as the orders become very large. Not all groups of large order are difficult to handle, however. Take for example IGI = 1, 000,000. It is a simple matter of using (2.9) on the possibilities for n5 that proves G is not simple. Nonetheless, when the larger orders are difficult, they can be very difficult. They are generally more cumbersome when their orders are comprised of quite a few primes close in size. It is no wonder that Marshall Hall, Jr. employed computer assistance in the course of his completion of the range problem up to order one million. 29
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3. The Simple Groups 3.1 Infinite Families of Simple Groups 3.1.1 The Alternating Groups "I have often in my life ventured to advance propositions of which I was uncertain; . it is too much to my interest not to deceive myself that I have been suspect of announcing theorems of which I had not the complete determination ... subsequently there will be, I hope, some people who will find it to their profit to decipher all this mess." (Galois [15]) The history of group theory itself begins with the discovery of the first compositely ordered simple group, A5 The process that led to the discovery of this simple group actually led to the idea of the study of group theory. It began with Evariste Galois (18111832) who led a very short but mathematically productive life, although it took time and scrutiny for anyone to understand his ideas. The above quotation was on the final page written by Galois before he died "for so trivial a thing" [16] in a duel when he was twenty one years of age. Many of the terms that he used were not rigorously defined, and his results were not often proven, being hurriedly jotted on a piece of paper. Yet Galois did have the first concept of groups as we define them today, and used them somewhat abstractly in his studies of solvable polynomials. Galois was working on the popular algebra problem of the eighteenth and into the nineteenth centuries, the factorability of polynomials over a field F. 30
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Galois' approach to this problem is rooted in the workings of permutations. The possible roots of a polynomial of degree n can be permuted in n! different ways. For example, look at the fourth order polynomial in the complex field: f(x) = (x2 + l)(x2 3). The four roots of the polynomial are x = i, i, V3, and V3. Suppose we let f3 = i I=V3 6 = V3. Then we have permutations of these four letters such as R1 = ( a f3 1 6 ) f3a16 which switches a and f3 and leaves the other two fixed. There are 4! = 24 similar permutations. A subgroup of the group of 24 permutations can be formed in the following way. Look at any polynomial equations involving a, /3, 1, or 6. Some equations express a true statement if the numerical values of a, (3, 1, or 6 are substituted, and some do not. For example, the equation 123 = 0 is true for 1 = V3, as is a+ f3 = 0 for the given values of a and f3. An equation such as 2/36 = 2 is obviously not true. The group of permutations which preserve the truth of the true equations form a subgroup of the permutation group. Notice that any true equation remains true if a and f3 are interchanged, and similarly if 1 and 6 are interchanged. Galois called this subgroup of permutations the group of the equation, G. In our example, this group consists 31
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of R1 = ( ; : : : } R, = ( : : ; } R, = ( ; : ; ) ,I = ( : : : : } The first concept of a normal subgoup was born by examining the group G. Choose a polynomial expression, w, which is rational in the roots of our original equation but has the following property: it's numerical value, t, stays fixed for some elements of G, but changes for others. Then those elements of G which fix t form a subgroup, H, of G. Galois showed that if tis a root of the (irreducible over F) binomial equation xPc = 0 where pis prime, then the subgroup His in fact normal in G. This process continues to reveal a method of solving equations by radicals, and also the inspiration for studying simple groups. Form a new field F(t) which is the smallest field containing both F and t. The subgroup H is then the group of the equation over the new field, F(t). Repeat the above process on H to find a normal subgroup of H, and a new field, F(t, t1 ) where t1 is the numerical value of the chosen expression. The process can be repeated until we are left with the identity permutation as the subgroup. In this case, the original equation is said to be solvable by radicals over the created field, F(t, t1 ... tn) Furthermore, we have a series of normal subgroups much like 1 = Ho
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that an equation was solvable if each index in the composition series was prime, and not solvable if some index was not prime. This is precisely what happens to quintic equations. Some composition factor in the composition series is compositely simple, not having prime order, and the end result of the identity permutation is never obtained. The simple group that was discovered by Galois by way of the in solvability of the quintic was the simple group of order 60. By 1832, Galois recognized this group as simple stating "The smallest group of permutations which an indecomposable group can have, when this number is not prime is 5 4 3 [15]." Galois stated this without proof and it wasn't until 1870 that Jordan would verify this result. In fact, Jordan gave better definition to the notion of a composition series which was only one great feat of his 1870 work, Traite des substitutions et des equations algebriques [15], which further inspired the study of simple groups. By this time, mathematicians were still concerned with the solution of algebraic equations, and this was the foremost purpose of the Traitf.. The use of permutation groups was still being explored and expanded, and groups were generally represented as such. Thus, Jordan discovered that the simple group of order 60 which was tied to the quintic equation was actually the alternating group on five letters, A5 An alternating group is the subgroup of the permutation group made up of all even permu tations. (A permutation is even if it can be written as a product of an even number of 2cycles, or transpositions.) Jordan went further than proving the simplicity of A5 He presented a (flawed) proof for the simplicity of all alternating groups, An, for n 5. This was the first infinite family of simple groups 33
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to be discovered. As an example of the permutation group theory used, the following is a proof for the simplicity of As: Theorem 3.1 As is simple. Proof: The cycle structures of the elements in As are the following: 1s, 1 22 12 3, and 5. This notation indicates that there are permutations which fix five letters (the identity), fix one letter and has two 2cycles, fix 2 letters and has one 3cycle, and which has one 5cycle. The orders of the elements in As which are made up of these cycle structures can be obtained by finding the least common multiple of the sizes of cycles for each type. That is, the order of the elements that are made up of two 2cycles and fix one point is 2 (LCM of 2 and 1), etc. as shown below: cycle structure order of elements number of elements 5 1 2 3 5 1 15 20 24. The last column above shows the number of elements of each order. These numbers are easily obtained by looking at the order of As. For example, the number of elements of order 3 is the number of elements of order 5 is etc. To show that As is simple, we shall proceed by contradiction. Suppose that As contains a normal subgroup, S, which is not the identity or As itself. The possible orders of S must divide 22 3 5. Suppose that 3 I lSI. Then S contains a Sylow 3subgroup of As and since S is normal and every two Sylow psubgroups are conjugate (2.14), S must contain all Sylow 34
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3subgroups. Thus, S contains all elements of order 3. There are 20 elements of order 3, so lSI > 20 (accounting for the identity). Also, 3 I lSI and ISIIIAsl, so lSI= 30. Now suppose that 5 I lSI. By the same argument as above, S contains all Sylow 5subgroups, and thus all 24 elements of order 5. So lSI > 24, thus lSI = 30. Since 30 is divisible by both 3 and 5, S must contain all elements of both orders, 20 + 24, but this is impossible if lSI = 30. So suppose lSI = 4. Then S would be a normal Sylow 4subgroup, and thus would be the unique Sylow 4subgroup. But there are 15 elements of order 2, so this is also impossible. Finally, suppose lSI = 2. Then IAut(S) I = 1 since cp(2) = 1. (2.13) Using the N/C theorem (7), NA5(S)/CA5(S) = 1 thus, NA5(S) = CA5(S). Since Sis normal in As, NA5(S) =As which implies that CA5(S) =As. This is not true, since a counterexample can be found easily as a 3cycle which does not commute with a product of two 2cycles. So none of the possibilities work, and our assumption must be incorrect. Therefore, As is simple. 3.1.2 Simple Groups of Lie Type The remainder of the infinite families of simple groups can be clas sified as Lie groups. These include the classical groups, the groups of type G2 the Chevalley groups (of types E4 E6 E7 and E8), the twisted groups (of types E6 and D4), the Suzuki groups and the Ree groups (of types G2 and F4). These groups arise as automorphism groups of corresponding simple Lie algebras. In general, since the theory of Lie algebras is too extensive for this 35
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paper, a Lie algebra is a vector space over a field with a product [X, Y] that is linear in both variables which also meets the following criteria: 1) [X, X] = 0 for all X in the vector space. 2) [[X, Y], Z] + [[Y, X], X]+ [[Z, X], Y] = 0 (the Jacobi identity) 3.1.2.1 The Classical Linear Groups It was Jordan again in his Traite who found the next four infinite fam ilies of simple groups, although he was not completely aware of the simplicity of each. Jordan obtained orders, generators and the factors of composition of some of these groups and was not explicit about the infinite families involved. We have seen how the simplicity of the infinite family PSL(m,pn) was finally proven by Dickson in 1897. In fact, Dickson worked on extending Jordan's results on all of the linear groups from 1897 to 1899. Dickson and Dieudonne are also credited with further investigating all of the linear groups in the years 1948 to 1958. The groups are now known as the projective special linear, the symplectic, the orthogonal, and the unitary groups. All four are collec tively called the classical linear groups. They are each groups of matrices. The construction of the first two are given below, and the construction of the orthogonal and unitary are similar in that they are each groups of invertible matrices factored out by the group's center. Projective special linear: The general linear group, G Ln ( q) is the group of all nonsingular linear operators of a vector space V where V has dimension n over the field of order q. Thus, GLn(q) is a group of n by n matrices. The order of GLn(q) can be given by the following: IGLn(q)l = (q1)qn(n1)/2(q21) ... (qn1). 36
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The subgroup of matrices with determinant 1 is normal and called the special n(n1) linear group, SLn(q). The order of SLn(q) is given by q2(q2 1) ... (qn 1). The center, Z, of GLn(q) consists of transformations of the form Tx = AX for A not 0. The center of SLn(q) can be denoted Z n SLn(q) and the factor group is the projective special linear group, PSLn(q). Its order is given by the following IPSLn(q)l = (n,:l)qn(nl)f2(q2 1) ... (qn1). Let the field be the Galois field GF(q) where q is a power of a prime. This group is simple for n 2 except for PS2(2) and PSL2(3). Let us look at a specific example of a projective special linear group. The simple group PSL3(2) is isomorphic to PSL2(7), both with order 168. We would construct PSL3(2) by looking first at G3(2) which consists of all nonsingular 3 by 3 matrices over the Galois field, GF(2). For example, 1 1 0 the matrix 1 0 1 is an element in G3(2). The order of G3(2) 1s 0 1 1 (2 1)23(23 1)(22 1) = 168. Note that this is the same as the order of PSL3(2) and indeed, they are isomorphic. The reason for this is that all matrices in G3(2) have determinant equal to 1 mod 2, thus all elements in G3(2) are also in S3(2). The center of S3(2) consists of the identity only, The construction of PSL2(7) isomorphic to PSL3(2) begins with G2(7), the group of nonsingular 2 by 2 matrices over GF(7). G2(7) has order 6 7 (72 1) = 2016. An element in this group looks something like 37
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[ : : ] or [ : : ] where the matriT entries are modulo 7. If we restrict ourselves to all matrices in G2(7) with determinant 1 mod 7, for example [ : : ] we have SL,(7) with order 336. The center of G2{7) consists of matrices like [ : ] [ : : ] ... [ : : ] The center of SL,(7) are those matrices in the center with determinant 1 mod 7, which are only 2, [ : ] and [ : : ] The simple group PSL2(7) is the factor group of SL2(7) and these two elements. Comparing the order with the formula given above, we see that IPS2(7)1 = 7 (72 1) = 168. Projective Symplectic: Suppose that the vector space V from above has a skewsymmetric, bilinear nonsingular scalar product so that (x, y) = (y, x), and (x, x) = 0. The symplectic group, Spn(q) where n =2m, consists of those linear transformations which preserve the above symplectic form. In particular, if A, B, e, and D are m x m matrices, then the transformation represented by the matriT [ ; ; ] is symplectic exactly when the following hold: AteetA = 0, AtDetB = I, and BtDDtB = 0 [5]. The projective symplectic group, PSpn(q), is the factor group Spn(q)/Z(Spn(q)) where Z(Spn(q)), the center of Spn(q) is made up of scalar matrices. PSpn(q) is simple except for PSp2(2), PSp2(3), and PSp4(2). The order of PSpn(q) is given by the following formula: (qm2 (q2m 1)(q2m2 1) ... (q2 1))/(q 1, 2). 38
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An example of a projective symplectic group is PSp2(9) which con tains 360 elements and is isomorphic to A6 Sp2(9) is a subgroup of GL2(9), the set of 2 x 2 matrices over the Galois field of 9 elements. First, we construct GF(9) by looking at the irreducible polynomial x2 + 1 over Z3 We find that GF(9) rv {ax+ b + (x2 + 1)} and the elements are {0, 1, 2, x, x + 1, x + 2, 2x, 2x + 1, 2x + 2}. Following the equations above, and simplifying our example by only looking at elements for which B = C = 0, we can write a couple of elements of Sp2(9): [ x + 1 0 ] and [ 2 x 0 ] The two elements in Z(Sp2(9)) are [ 2 0 ] 0 x+2 0 x 0 2 and [ : : ] and since PSp2(9) is the factor group of SP2(9) and these two elements, we know that [ x + 1 0 ] and [ 2 x 0 ] are also in PSp2(9). 0 x+2 0 x One can easily verify that (x + 1)(x + 2) = 1, and (2x)(x) = 1 in GF(9). 3.1.2.2 Other Lie Groups A brief mention of the history of other groups of Lie type is in order. During the period 1901 to 1905 a new family of simple groups of Lie type was discovered by Dickson. Until 1955, classical linear and this new family were the only simple groups of Lie type known. Claude Chevalley produced a much needed new way of approaching these simple groups and in the process, he discovered several more infinite families of simple groups of Lie type. These are referred to as the Chevalley groups. Chevalley's progress on the groups of 39
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Lie type successfully increased the interest in the field, and it wasn't long before new infinite families of simple groups of Lie type were found. In particular, in 1960 Suzuki discovered his infinite family while working on what is now called a classification problem (see Chapter 4). He was trying to find all simple groups in which the centralizer of an involution (that is all elements that commute with a particular element of order two) is a group of order 2n. In the process of trying to eliminate all possibilities except for PSL(2, 2n) and PSL(3, 2n), n 2, Suzuki found another family with the given property. These are Sz(2n). In 1961, Rhimak Ree was analyzing the Suzuki groups using a particular method (Steinberg's) which had produced infinite families of Lie type before, and came up with two additional families. Thus, the Chevalley, Steinberg, Suzuki, and Ree groups are the simple groups of Lie type along with the classical linear groups. 3.2 The Sporadic Simple Groups The remaining known simple groups do not fit into any large model of similar attributes as do the infinite families. They were discovered often one by one. Some do fit together by method of discovery or by construction. We will examine these properties briefly below. First, the following table lists the 26 sporadic simple groups, their order (if not too large), their discoverer (according to some references), and the date of their discovery. 40
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Name Order Discovered by Date Mu 24 32 5. 11 Mathieu 1895 M12 26 33 5. 11 Mathieu 1899 M22 27 32 5 7 11 Mathieu 1900 M23 27 32 5 7 11 23 Mathieu 1900 M24 210 33 5 7 11 23 Mathieu 1900 J1 23 3 5 7. 11 19 Janko 1966 J2(HaJ) 27 33 52 7 Hall, Janko 1967 J3(HJM) 27 35 5. 17. 19 J anko,Higman,McKay* 1969 HS 29 32 53 7 11 Higman,Sims 1968 MeL 27 36 53 7 11 McLaughlin 1969 Suz 213 37 52 7 11 13 Suzuki 1969 He 210 33 52 73 17 Held, Higman, McKay 1969 Co1 221 39 54 72 11 13 23 Conway, Leech* 1969 Co2 218 36 53 7 11 23 Conway* 1969 Co3 210 37 53 7 11 23 Conway* 1969 Fi22 217 39 52 7. 11. 23 Fischer 1969 Fi23 218 313 52 7 11 13 17 23 Fischer 1969 Fi;4 Fischer 1969 Ly Lyons,Sims 1971 Ru 214 33 53 7 13 29 Rudvalis, Conway, Wales 1972 O'N 29 34 5 73 11 19 31 O'Nan,Sims 1973 M Fischer 1974 B Fischer 1974 F3 215 310 53 72 13 19 31 Thompson,Smith* 1974 F5 214 36 56 7. 11 19 Fischer,Smith, Harada* 1974 J4 Janko* 1975 Table 3.2: The Sporadic Simple Groups The names of discoverers followed by a star are those which have some discrepancy depending on sources. Those orders denoted by a star are 41
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too large to fit this table. For example, the order of the group M, the largest of the sporadic simple groups is 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 3.2.1 The Mathieu Groups The Mathieu groups, M11 M12 M22 M23 and M24 are the earliest sporadic simple groups to be discovered. They were described by Emile Math ieu (18351900) in 1861 and 1873. Mathieu was influenced by Cauchy's work on permutations. Mathieu was investigating multiply transitive functions, and thus permutation groups and multiply transitive permutation groups. A permutation group on a set A is said to be ntransitive if for any ordered pair of ntuples of elements of A, there exits some element of the permutation group that maps one tuple to the other. That is, xig = Yi for 1 ::; i ::; n where Xi and Yi E A and g E permutation group of A. A transitive function is one which is left invariant under the permutations of a transitive group, which was discovered by Mathieu. In the course of his work, Mathieu attempted to 'extend' transitivity by constructing an ntransitive permutation group out of a (n1)transitive permutation group. He was able to find an algorithm for the construction of these groups, when their construction was possible. The high est transitivity found in a simple group is 5transitive, and Mathieu discovered the 5transitive permutation groups on 12 symbols and on 24 symbols which are M12 and M24 The other Mathieu groups arose as subgroups of these and a subgroup of M23 For example, M11 is the subgroup of M12 formed as the 42
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stabalizer of a point in M12. Each of the Mathieu groups are multiply transi tive. The simplicity and uniqueness of the Mathieu groups was not expressed until the 1930's in a paper by Witt who was describing what is called the Steiner system. The Mathieu groups are now normally described in terms of this system. 3.2.2 Centralizer of Involution Problems The next sporadic simple groups were not discovered until around one hundred years after Mathieu's find. The first of these is Janko's first, J1 and the method by which it was discovered became an important part of the theory of simple groups and an important method to discover other simple groups. The central feature of the method is the centalizer of an involution, or the centralizer of an element of order two. We have seen how centralizer of involution questions led to Suzuki's infinite families of simple groups of Lie type. The centralizer of an involution as an entity is important due to a number of results. Two of these are a theorem due to Brauer and Fowler and the FeitThompson Theorem (or Odd Order theorem), which are both looked at below. Because an involution is an element of order two, the order of a group containing an involution must be divisible by two. If it were guaranteed that a simple group contained an involution, this may increase the potential of classifying simple groups according to something related to involutions. This result was indeed obtained, in the FeitThompson Theorem. It was not a swift theorem to come up with however, and the odd versus even order of simple 43
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groups was a long standing question. In fact, conjectures on this question date back to 1895 and Burnside. Burnside had a good hunch that simple groups must necessarily have even order, and from 1895 to 1901, he attempted to show this. He was successful at proving that all simple groups with orders under 40,000 had even orders, yet he could not generalize his result. He believed that the necessary technique to prove his conjecture was character theory. The problem came alive again in 1957 with the work of Suzuki who was indeed using character theory. Suzuki was able to prove that any simple group in which the centralizer of any element (other than the identity) was abelian has even order. This result was extended in 1960 by Feit, Hall, and Thompson who proved that a simple group must have even order if the centralizer of any nonidentity element is nilpotent, i.e., all of its Sylow subgroups are normal. Three years later, the same two, Feit and Thompson, took 255 pages of the Pacific Journal of Mathematics [6] to prove that all groups of odd order are solvable. This means that the composition series of a group of odd order contains composition factors of prime order, which indicates that the group is not simple. Thus, any simple group must have even order, and therefore, must contain involutions. A result pertaining directly to the centralizers of involutions was actually found earlier than the FeitThompson Theorem. In 1954, Brauer and Fowler proved that there are at most a finite number of simple groups in which the centralizer of an involution has a given structure: Theorem 3.2 IfG is a finite simple group of even order, andt is an involution in G, then IGI ::S: (ICa(t)l2)!. 44
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Ca(t) denotes the centralizer in G oft. Since there can only be a finite number of groups with orders less than a particular number, then there are only a finite number of groups with the centralizer of an involution isomorphic to a given centralizer. This provided the idea of at least some classification of finite simple groups by the structure of the centralizer of involutions. The importance of this result was furthered by the FeitThompson Theorem since then the result pertained to all simple groups, not just simple groups of even order. This theorem has been improved upon in the more recent years in many variations using the idea of a central involution which is an involution in the center of a Sylow 2subgroup. In general, it has been established that if a centralizer of a central involution in a questionable simple group is isomorphic to the centralizer of a central involution in a known simple group, then the two simple groups are isomorphic. These are powerful results which may allow for the characterization of a simple group by its centralizer of a central involution. An example of this type of theorem is the following due to Brauer: Theorem 3.3 Let G be a simple group which contains an involution whose centralizer is isomorphic to GL2(q) factored by a subgroup of odd order in the center of G L2 ( q), and where q is an odd prime power congruent to 1 mod 4. Then either 1. G rv PSL3(q), or 2. G rv M11 and q = 3. There are many other such theorems, and the theory involved in the study of centralizers of involutions is extensive. This paper will only be able to concern itself with a brief description of the discovery of some of the sporadic simple 45
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groups due to centralizer of involution theory. Let us return to the next sporadic simple group to be discovered, J1 The story of Janko's first group begins with the centralizer of the involutions in one family of Ree groups of Lie type, denoted R(3n). It was found that the centralizer of an involution of R(3n) is isomorphic to the group Z2 x PSL2(3n), the external direct product. It was also noted that the Sylow 2subgroups are elementary abelian of order 8. Thus, an interesting task became to determine all simple groups with Sylow 2subgroups with the above properties which have centralizers of involutions isomorphic to Z2 x PSL2(pn), pan odd prime. For pn = 5, the new simple group J1 was discovered. Janko proved the following theorem: Theorem 3.4 If G is a simple group with abelian Sylow 2subgroups of order 8 and the centralizer of an involution of G is isomorphic to Z2 x PSL2(5), then G is a uniquely determined simple group of order 175,560. Moreover, G is isomorphic to the subgroup of G L7(11) generated by the following two elements of order 7 and 5: 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 s1 = o o o o 1 o o 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 3 2 2 1 1 1 3 1 3 1 3 1 3 3 1 1 3 1 3 3 2 and S2= 1 3 1 3 3 2 1 3 1 3 3 2 1 3 3 3 3 2 1 2 1 1 1 1 1 3 3 1 Janko was the lucky receptor of further inspiration which led to two 46
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other sporadic simple groups, J2 and J3 After the discovery of J1 Janko looked further for possible centralizers of involutions inspired by those in the Mathieu groups. He tried a centralizer of an involution which was isomorphic to the extension of a group of order 32 by A5 He actually found two new groups with the same centralizer of an involution, J2 and J3 Hall and Wales proved the existence of J2 and Higman and McKay proved the existence of J3. The question of the existence of two simple groups with isomorphic centralizers of involutions led to the discovery of the next sporadic simple group in the story. (We now cease chronological order). D. Held knew that the groups M24 and PSL5(2) have involutions with isomorphic centralizers. While investigating this phenomena, Held discovered yet another simple group with the same centralizer of an involution, He. This is the only case of three simple groups with isomorphic centralizers of involutions. The next sporadic simple group to be obtained by examining cen tralizers of involutions is Ly. John McLaughlin's group, M c, to be discussed below, has a centralizer of an involution which is isomorphic to the group As which denotes the perfect extension of As by Z2 The idea then arose to study centralizers of involutions which are isomorphic to An for n 5. On such an investigation, Richard Lyons, who was a student of Thompson's, made the following discovery: Theorem 3.5 If G is a simple group in which the centralizer of an involution is isomorphic to An, n = 10 or 11, then n = 11, and 47
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In fact, the result was shown that simple groups could only arise from central izers of involutions isomorphic to A8 and A11. Incidentally, It was Janko who had worked on this problem. He showed that when n = 9 and 10, there were no simple groups with the said centralizer of an involution. He gave up before working on n = 11. He did however discover the last sporadic simple group falling under the category of centralizer of involution problems, and that was J4. 3.2.3 Rank 3 Permutation Groups The group J2 has a structure which became important to the con struction of four more sporadic simple groups. J2 is said to be a primitive rank 3 permutation group. A group G has permutation rank r if G is transitive on a set D and the subgroup of G that fixes a point of D has exactly r orbits on D. Recall that a group is transitive if for a set D, and any two elements a and f3 E D, there exists an element g E G such that a g = {3. Also, the subgroup of G that fixes a point of n are those elements in G for which a g =a. The orbits on n are sets of the form {a. g I g E G} n. The orbits of n partition D. The group J2 fits this description if one considers the maximal subgroup of index 100, H J2 The permutation representation of J2 on the right cosets of H is a transitive action which produces a primitive permutation representation of J2 of degree 100 (i.e., J2 is a transitive permutation group of degree 100), which takes the role of the set D above. On this set, H fixes one point and its action produces two other orbits, rendering J2 a rank 3 permutation 48
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group. In fact, the existence of J2 was proven using the theory of rank 3 permutation groups. The maximal subgroup H happened to be isomorphic to the simple projective special unitary group, U3(3). Donald Higman and C. Sims noted the similarity in permutation properties between the groups U3(3) and M22 and in record time were able to construct a new simple primitive rank 3 permutation group using M22 as the maximal subgroup and 'extending' it to obtain the group HS. A similar technique was used by McLaughlin who started with the group U4(3) to extend it to a rank 3 permutation group that is simple, called M cL. Suzuki obtained his sporadic simple group Su in the same manner starting with G2(4) (a Chevalley Lie type simple group). Finally, the fourth rank 3 permutation group was constructed by Rudvalis using the Ree group, 2 F4(2). It is Ru. 3.2.4 The Remaining Sporadic Simple Groups This shall serve to briefly describe the discovery of the remaining sporadic simple groups. The Conway groups, Co1 Co2 and Co3 came out of the study of an automorphism group of a lattice called the Leech lattice which is determined by a set of vectors in 24dimensional Euclidean space with integral coordinates. The three simple groups happen to have been sub groups of this automorphism group. The Fischer groups, Fi22 Fi23 and were discovered by Fischer while studying classes of 3transpositions. These are conjugacy classes generated by involutions such that the product of two involutions in the class has order 2 or 3. Fischer generated groups by these classes, and put further conditions on the groups proving that the new group 49
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is either a symmetric group, a certain classical group, or one of the three Fis cher groups. Fischer then turned to groups generated by {3,4}transpositions (two involutions in a class have a product of order 2, 3, or 4). Two groups, B and M, or Baby Monster and Monster were discovered. The Monster is the largest sporadic simple group, and a representation for it was obtained by hand by Robert Griess. It was in terms of square matrices that were 196,883 by 196,883 in size. The groups B, F3 and F5 are actually subgroups of the Monster. F3 was found by Thompson, and F5 is attributed to Harada, Norton, and Smith. The O'Nan group came out of the study of groups with particular Sylow 2subgroup structure. The methods used to discover new sporadic simple groups were often haphazard, as Daniel Gorenstein says, "some of the groups seemed literally plucked from thin air [9]." Sometimes the techniques used were character theorybased. In fact, Feit, Thompson, and Brauer were quite well known for their work in and development of character theory. There are really three phases in determining a new simple group, and only one of those phases I have taken consideration of here. There is the discovery, which is what I have described, there is the existence and there is the uniqueness. Often several different individuals contribute to the determination of the existence and the uniqueness of a new simple group. The discoverer is generally who the group is named after. The simple groups found later than J1 had the timely advantage of computers to aid in their discovery, existence, and uniqueness questions. 50
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4. The Classification Theorem 4.1 History The study of centralizers of involutions proved not only very useful in locating certain sporadic groups, but also marks what some would consider the start of the classification project. As noted above, in 1954 Brauer made his great discovery that there are only a finite number of groups with their centralizers of an involution having a particular structure. This seemed to spur the idea of the characterization of simple groups according to their centralizer of an involution. It was in fact Brauer who suggested such a thing, and was successful with his use of character theory in certain cases. Others contributed to this line of study, and some good results were obtained, often with the discovery of sporadic simple groups. Brauer's ideas served to provide a new avenue down which some could dream of an overall classification of all finite simple groups. There were also certain advances in theory that inspired many to take part in the study of simple groups. The work of Brauer and Suzuki in character theory provided one. The new discoveries about Lie groups in the 1950's is another. But in the 50's, there was still much to be accomplished before a classification idea could become a reality. The 1960's provided the study of simple groups with some of those high powered results it needed. The most influential is the famous 1963 Feit Thompson theorem, or the odd order theorem, which states that all groups of 51
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odd order are solvable. It was not only the result that was terribly influential, but also the structure of this 255 page proof. Thompson was also respon sible for another very important result which took 410 pages and six years (19681974) to complete. This is the classification of minimal simple groups, or those simple groups which have only solvable groups as subgroups. Fol lowing Brauer's program, Suzuki was able to characterize all simple groups in which the centralizer of an involution has a normal Sylow 2subgroup in 1965. Sylow 2subgroups were becoming as telling as centralizers of involutions, and many results stemmed from their study. In particular, Gorenstein and Walter characterized simple groups with dihedral Sylow 2subgroups also in 1965. In 1969, Walters classified simple groups with abelian Sylow 2subgroups. These are general characterizations. An example of a specific characterization is Glauberman's Z* theorem of 1966 which showed that every involution is conjugate to another involution in its centralizer. These are only a few of the important steps taken in the 1960's, and many other results were to follow. By the 1970's, there were many roads to classification, although no systematic idea of its achievement. There were also many sporadic groups discovered in the 1960's, and some wondered if there was an endless supply of them. Thus, in 1972 at the University of Chicago, when Gorenstein presented his idea of a 16step plan to classify all simple groups, not many were opti mistic. Gorenstein projected that to complete his program would take about thirty years. The task seemed daunting, yet a few tackled portions of the plan. The project was propelled rather suddenly by a newcomer, Michael As chbacher, who "came on now like a whirlwind, moving directly to a leadership 52
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position and sweeping aside all obstacles, as he proved one astonishing result after another [9]." The results being made at this time were obviously highly complex, and therefore cannot be handled in this paper. It should be noted that the original plan of 30 years was decreased to an actual 10 years, and Gorenstein attributes this to Aschbacher. The completion of the classification theorem took place in January of 1981. Some Methods The methods used in pursuing the idea of the complete classification of finite simple groups naturally changed as progress was made. As can be noted from previous chapters, character theory was used frequently for many results. It turns out that character theory is limited in handling 'large' simple groups. Smaller groups, such as lower ordered groups, or groups with Sylow 2subgroups that are restricted structurally (such as abelian) are good candi dates for the use of character theory in examining them closely. However, as the questions about the group's internal structure became more broad, new techniques were needed. These techniques are called local grouptheoretic analysis, or local analysis. It was the FeitThompson theorem that initiated the practice of local analysis. The predecessors of the FeitThompson theorem, Suzuki's abelian centralizer result and the Feit, Thompson and Hall result on nilpotent centralizers (see p.45) used character theory to develop the lattice of proper subgroups of the group in question. This required analysis of ev ery subgroup. This process could not be used in the FeitThompson theorem, since there was no information on the structure of centralizers to rely on. A new set of techniques was developed by Thompson, and their main emphasis 53
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was to look at centralizers and normalizers of prime power order subgroups and analyze their relationships. A new term was coined for the normalizer of a nonidentity prime power subgroup, and that was local or plocal subgroup (p being the prime power). Thus, the techniques of local analysis are the methods of examining local subgroups. The local analytic methods were explored further by Thompson in his classification of minimal simple groups, and his Ngroup theorem of 1968. An Ngroup is a simple group whose local subgroups are each solvable. Thompson explored all possible simple groups fitting this description, and was able to classify the Ngroups: Theorem 4.1 If G is a simple N group, then G is isomorphic to one of the following groups: PSL2(q), where q > 3, Sz(q), where q = 22n+I, n 1, PSL3(3), U3(3), 2 F4(2)', A7, or Mn. U3(3) is a unitary group, and 2 F4(2)' is a Ree group of Lie type. Thompson's strategy was to show that an arbitrary Ngroup has internal structure that looks like one of the groups listed. Then resemblance was shown to be actual isomorphism. This process is mirrored in the classification theorem, as will be seen below. One concept that was invaluable to Thompson's Ngroup theorem and later to local analysis in general was the idea of 'embedded subgroups'. An example of a type of embedded subgroup is a 'strongly embedded subgroup' M of G This means that IMI is even and the following hold: 54
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1. C0(t) M for every involution t of M 2. N0(S) M for each Sylow 2subgroup S of M. Strongly embedded subgroups themselves were actually classified by Bender in 1971 as either PSL2(q), U3(q), or Sz(q) for q even. While local analysis was developed, and results of a different nature were obtained because of the change of emphasis, there were also further changes in direction by creative individuals. A couple of these different ap proaches are mentioned now. The method of both the FeitThompson theorem and many classification theorems that followed was generally to look at min imal counterexamples and either derive a contradiction to the theorem state ment, or show isomorphism of the group in question to a known simple group. The procedure to achieve this was to examine relatively small subgroups to develop the local subgroup structure. Helmut Bender changed this approach in his attempt to simplify the proof of the FeitThompson theorem. He stud ied the intersections of maximal subgroups which contained the centralizer of some involution. This approach is called the Bender method, and was used to dramatically reduce the complexity of such theorems as Walter's result about abelian Sylow 2subgroups, and Gorenstein's and Walter's result about dihe dral Sylow 2subgroups. Originally, Bender was looking for a revision of the classification as a whole, beginning with the FeitThompson theorem, but his method became a useful tool in itself. Another innovation that was second only to local analysis techniques 55
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was Fischer's internal geometric analysis. We have seen the work of Fis cher with respect to the discovery of sporadic simple groups. His ideas of 3transpositions went much further than only the discovery of his sporadic groups however. Recall that a class of 3transpositions is a conjugacy class of involutions where the product of any two has order 1, 2, or 3. Also, the group G in question is generated by this conjugacy class. Fischer's geometric approach was to consider a graph whose vertices are the elements of the conjugacy class, and any two elements which commute with each other are connected by an edge. The group G acts as a group of automorphisms of the graph since under conjugation, G permutes the vertices of the graph but preserves the incidence relation on the graph. Thus, Fischer saw that the structure of the group G is related to the geometry of the graph. His work inspired others such as As chbacher, and the definitions of connected and nonconnected came from the nature of the graphs. We will see that these play a very important role in the classification theorem. 4.2 The Theorem The entire classification theorem is a monumental enterprise of be tween 10,000 and 15,000 pages, taken from around 100 contributors, and written over a period of more than 30 years. There are articles stretching out among perhaps 500 journals that comprise the theorem. The main contrib utors are an international group mainly from the U.S., Germany, England, Canada, Australia, and Japan. Results were collected starting around the late 1940's and complete classification was obtained in the early 1980's. We 56
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have seen that a systematic approach to the classification was proposed as late as 1972. The theorem itself states that all finite simple groups have been found. That is, any finite simple group is isomorphic to one of those already discovered: Theorem 4.2 Main Classification Theorem: Every (nontrivial) finite simple group is isomorphic to one of the following: 1. A group of Lie type 2.An alternating group 3. One of the above mentioned 26 sporadic groups (see p.42). The general structure of the theorem is that of induction. A minimal simple counterexample G is chosen such that G is assumed not isomorphic to any known simple group, and any simple group with order less than G is a known simple group. Also, suppose that the group G has a set of properties, X. Given this information, one can prove that G is actually a known simple group, deriving a contradiction. The inductive nature of the proof is important for looking at internal properties of subgroups of G. For example, there is a result which states that: Theorem 4.3 Given a minimal simple counterexample G with a set of prop erties X, if H c G and K
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is a known simple group. Suppose that we have a minimal counterexample G with X properties that we assume is not a known simple group. This assumption forces us to consider the internal properties of G to be as complicated as any finite group. We cannot assume that G looks like a known simple group from the start, for that is what we are trying to prove. The next step is to force our counterexample to look like a known simple group. Obviously, this is not an easily accomplished task and many of the high powered local analysis techniques must be used carefully in the examination of the internal structure of the group. There are many possibilities for a group G with X properties, and each must be considered. This accounts for much of the complexity and length of the theorem, since there are around 100 different paths to follow to show that G looks like a known simple group. The paths themselves are determined by the properties of G, so each case is different. The classification theorem is complete in that it exhausts all of the possible structures of G and leads all possible simple groups to the structure of a known simple group. In order for us to know that our simple group looks like a known simple group, we must have a very detailed description of the known simple groups. This part of the theorem is called the recognition theorems. Once it is determined that G 'looks like' a known simple group, then the steps toward isomorphism must be taken. That is, internal resemblance must be shown to be actual isomorphism. It is to be noted that the structure of the classification theorem is very similar to that of the FeitThompson theorem. In fact, one can break 58
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down the process of both into three steps: [10] 1.) Use the given properties of G to determine the structure and embedding of maximal subgroups containing or intersecting centralizers of involutions by local analysis. 2.) Eliminate as many of these possible configurations by using char acter theory on smaller groups, local analysis on larger groups, and arithmetic methods. 3.) Use recognition theorems (generators and relations) to prove that the only possible configuration left is isomorphic to a known simple group. Beginning with the last step first, each of the known simple groups must be recognizable by some defining feature. These recognition theorems usually are in terms of generators and relations, especially for the groups of Lie type. The alternating groups can also be characterized by generators and relations, as the following theorem shows: Theorem 4.5 If the group G is generated by the elements x1 x2 ... Xn_2 subject only to the relations xf = 1, x7 = 1 for 2 i n2, (xixi+1)3 = 1 for 1 i n3, and (xixj)2 = 1 for 1 i n4 and i + 1 < j, then G rv An. The recognition theorems for the sporadic groups usually depend on how the sporadic group was constructed. For example, those sporadic groups which were constructed by their centralizer of an involution can be character ized by this centralizer. (Theorems 3.4 and 3.5 are examples of recognition theorems). Those sporadic groups which are rank 3 are characterized by their one point stabilizers. (See p.49) Thus, much of the discussion in Chapter two serves to describe some of the recognition theorems. If the counterexample 59
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group G* is shown to have such characteristics as are given in the recognition theorem of group G, then the purpose of the recognitions theorems is to state that G* is necessarily isomorphic to G. The first two steps of the classification theorem are then to prove that G* has some defining features that are in one recognition theorem. We have seen that centralizers of involutions and Sylow 2subgroups play an important role in the internal structure of any simple group. Many sophisticated features of a group have been discovered in relation to these two. One of the reasons for this is that Sylow 2subgroup structure depends on the properties of centraliz ers of involutions (since Sylow 2subgroups contain all of a groups' involutions, and there is always an involution in the center of a Sylow 2subgroup), and centralizers of involutions can often lead to recognition theorems. There are complicated techniques to achieve this leap however, including what are called 'fusion arguments'. The purpose of this line of theory is to give precise descrip tions of the way in which involutions in a Sylow 2subgroup are conjugate in the group. Some of the famous results are Glauberman's Z* theorem, Thompson's fusion lemma, and Alperin's fusion theorem. Embedding is another property of subgroups which developed into important theory. What are called 'sig nalizer functor methods' grew out of the study of embedded subgroups. The accumulation of all of the possible internal structures of a simple group can be summarized in the four part division of the main classification theorem proof: The classification of nonconnected simple groups, The classification of connected simple groups of component type, The classification of 'small' simple groups of characteristic 2 type 60
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The classification of 'large' simple groups of characteristic 2 type [10]. The definitions of each of these are quite involved. We have seen how connected and nonconnected groups might arise. Let us now define character istic 2 type. Definition X is characteristic 2 type if F*(H) is a 2group for every 2local subgroup H of X. Now F* (H) is called the generalized fitting subgroup of X and F*(H) = L(X)F(X). F(X) is the fitting subgroup of X, which means it is the unique largest nilpotent subgroup of X. L(X) is the layer of X, which means that L(X) is the product of all subnormal quasisimple subgroups of X, with L(X) = 1 if no subnormal subgroups exist. A subnormal subgroup of X is a subgroup Y such that Y = Y1
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to the main objective and some methods of the proof. A revision of the proof has been suggested, and begun. It was spearheaded by Daniel Gorenstein who unfortunately died in 1992. With such a large proof to begin with, it is generally held that completely new techniques would have to be obtained before any remarkable reduction in length could be realized. When the theorem was nearing completion, a headline in the New York Times read, "A School of Theorists Works Itself Out of a Job", 1980. Yet all of those involved in the proof had positive ideas of the future of group theory. Gorenstein cited applications to such fields as mathematical logic and number theory due to the classification theorem [11]. The relationship between finite group theory and finite geometries was mentioned by Aschbacher as possibly benefitting from the classification theorem [1]. Also even within the field of group theory, many felt there was much to do. As Gorenstein comments, "the obituary for finite group theory has been totally premature [9]." The theorem itself is a testament to the perseverance and cooperative nature of human kind. It has been said in reference to the length and complexity of the theorem that either they have been a bit dim in finding the most effective techniques to prove the classification theorem, or they have been very clever indeed. 62
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REFERENCES [1] M. AsCHBACHER, The Finite Simple Groups and Their Classification, Yale University Press, 1980. [2] , Finite Group Theory, Cambridge University Press, 1986. [3] R. W. CARTER, Simple Groups of Lie Type, John Wiley and Sons, 1989. [4] M. J. COLLINS, Finite Simple Groups II, Academic Press Inc., 1980. [5] J. CONWAY, R. CURTIS, S. NORTON, R. PARKER, AND R. WILSON, Atlas of Finite Groups, Clarendon Press Oxford, 1985. [6] J. GALLIAN, The Search for Finite Simple Groups, Mathematics Magazine, 49 (1976), pp. 163179. [7] ,Contemporary Abstract Algebra, DC Heath and Co., 1994. [8] D. GORENSTEIN, Finite Simple Groups and Their Classification, Israel Journal of Mathematics, 19 (1974), pp. 566. [9] , Finite Simple Groups, An Introduction to their Classification, Plenum Press, NY, 1982. [10] , Classification of Finite Simple Groups Vol I, Plenum Press, NY, 1983. [11] , The Enormous Theorem, Scientific American, 253 (1985), pp. 104115. [12] M. HALL JR., A Search for Simple Groups of Order Less than One Million, in Computational Problems in Abstract Algebra, J. Leech, ed., Pergamon Press, NY, 1969, pp. 137168. [13] I. M. IsAACS, Algebra A Graduate Course, Brooks/Cole Publishing Co, 1994. 63
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[14] M. B. POWELL AND G. HIGMAN, Finite Simple Groups, Academic Press Inc., 1971. [15] R. SILVESTRI, Simple Groups of Finite Order in the Nineteenth Century, Archive for the History of Exact Sciences, 20 (1979), pp. 313356. [16] I. STEWART, Galois Theory, Chapman and Hall, 1989. [17] H. WEYL, The Classical Groups, Princeton University Press, 1946. 64
