A HISTORY OF FINITE SIMPLE GROUPS
by
Faun C C Doherty
B A Ob erlin College OH
A thesis submitted to the
University of Colorado at Denver
in partial ful llment
of the requirements for the degree of
Master of Science
Applied Mathematics
This thesis for the Master of Science
degree by
Faun C C Doherty
has b een approved
by
J Richard Lundgren
William E Cherowitzo
Stanley E Payne Date
Doherty Faun C C M S Applied Mathematics
A History of Finite Simple Groups
Thesis directed by Asso ciate Professor J Richard Lundgren
ABSTRACT
A group is a set together with an asso ciative binary op eration such
that there exists an identity element for the set and an inverse for each element
in the set All nite groups can b e broken down into a series of nite simple
groups which have b een called the building blo cks of nite groups The
history of nite simple groups originates in the s with Evariste Galois and
the solution of fth degree p olynomial equations In the twentieth century the
recognition of the imp ortance of nite simple groups inspired a huge e ort to
nd all nite simple groups This classi cation pro ject was completed in
We shall b egin by taking a historical lo ok at the earliest metho ds of analyzing
the structure of nite groups according to their order Finite simple groups
can b e divided into two typ es those b elonging to in nite families and the
sp oradic simple groups We shall lo ok at the discovery and representation of
many of these Finally we shall discuss the monumental to page
pro of of the classi cation of all nite simple groups
This abstract accurately represents the content of the candidate s
thesis I recommend its publication
Signed
J Richard Lundgren iii
ACKNOWLEDGEMENTS
I would like to thank Professor Lundgren for his supp ort in writing
this thesis Also thanks to my parents for their example and Michael for his patience
CONTENTS
Chapter
Intro duction
The Range Problem
Intro duction to the Problem
Sylow s Theorems
Other Theorems Corollaries Etc That Will Prove
Useful
Some History
H older
Cole Burnside
The Completion of the Range Problem Through Order
One Million
Some Examples
The Simple Groups
In nite Families of Simple Groups
The Alternating Groups
Simple Groups of Lie Typ e
The Classical Linear Groups
Other Lie Groups
The Sp oradic Simple Groups v
The Mathieu Groups
Centralizer of Involution Problems
Rank Permutation Groups
The Remaining Sp oradic Simple Groups
The Classi cation Theorem
History
The Theorem
References vi
Intro duction
Some have referred to the study of simple groups as the El Dorado
of nite group theory It has b een a very active eld of study through the
twentieth century and has its ro ots in the nineteenth as do es group theory
itself A group is de ned as a set together with an asso ciative binary op eration
de ned such that there exist an identity element for the set and inverses for
each element of the set The set is closed under the op eration A normal
subgroup H of a group G is a subgroup such that aH H a for all a G
1
Another de nition of normal is that a H a H for a 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 pap er the term simple group will refer to nite nontrivial
simple groups Simple groups are sp ecial kinds of groups that are the building
blo cks of all other groups thus the imp ortance in their study This idea was
recognized as early as by Evariste Galois and later a search for the
simple groups to ok place In the twentieth century this search culminated in
a monumental theorem which classi es all simple groups One of the earliest
metho ds of lo cating 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 pap er will outline the history of this
problem and the metho ds used through the analysis of groups up to order one
million The second chapter will describ e the simple groups by their typ es
in nite families of simple groups and the sp oradic simple groups How some
of these groups can b e represented as well as the metho ds of their discovery
will b e discussed Finally a general outline of the classi cation theorem will
b e given in the last chapter
The Range Problem
Intro duction to the Problem
Among the metho ds of determining all nite simple groups the ap
proach of examining individual groups of certain orders can seem at times slow
and metho dical Yet this task b egun in by Otto H older 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 up on 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 when Marshall Hall Jr completed the individual
examination of groups with particular orders through the order of
Ab out eleven individuals from to participated in the solution of this
problem each aided by the work and discoveries of those who came b efore
The range problem itself is not di cult to understand in light of the
search for simple groups It is simply this given a particular natural numb er
say n what can we say ab out the structure of any group having n elements
And in particular can we determine if the group has any normal subgroups
b esides 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 metho ds used to explore the structure of groups with a
given order One was to use the Sylow theorems and the other was to employ
character theory It will b e the task of this pap er to concentrate only on the
Sylow theorem metho ds thus a word ab out these theorems is in order
Sylow s Theorems
Ludvig Sylow a Norwegian mathematician came up with the Sylow
theorems in by way of the study of p ermutation group theory These
results lost no imp ortance with the development of abstract group theory in
fact their imp ortance grew The Sylow theorems as we state and prove them
to day are based on the fundamental concept known as Lagrange s theorem
and it is here that we shall start
Theorem Lagrange s theorem Suppose H G is a subgroup Then
jGj jH j jG H j
Note that jG H j is the index of H in G or the numb er of distinct
right cosets of H in G A right coset is the set H g fhg j h H g 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 numb er 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 numb er of distinct right cosets that
partition G
k
Theorem Sylow s Theorem If p j jGj then G has at least one
k
subgroup of order p for any prime p
Thus if any p ower of a prime divides the order of our group then the
group has a subgroup of order that p ower of the prime
k
Theorem Sylow s Theorem If H G and jH j p then H is
contained in some Sylow p subgroup
A Sylow p subgroup is a subgroup of G such that its order is equal
to the full p ower of p in the order of G For example if we have a group of
4 2 4
order a sylow subgroup would have order The set of all Sylow
p subgroups of G is denoted S y l G We know from the rst Sylow theorem
p
that S y l G is not empty We can also nd a Corollary b elow which
p
states that if only one Sylow p subgroup exists then it is normal in G This
fact will allow us to eliminate easily many integers as p ossible orders of simple
groups
Theorem Sylow s Theorem The number of Sylow p subgroups of G
i e jS y l G j written n has the fol lowing properties
p p
n mod p
p
e
and n mod p
p
e
if p jS S T j for al l S and T S y l G with S T
p
The examination of the structure of groups with a given order is feasible b e
cause of a numb er of other results b esides the Sylow theorems although many
of these results are based on the Sylow theorems A numb er of these results
shall b e listed b elow and referred to throughout this chapter
Other Theorems Corollaries Etc That Will Prove Useful
Theorem A nontrivial nite p group has a nontrivial center
A p group where p is prime is de ned as a group in which every element has
order a p ower of p The center of a group Z G is a normal subgroup of G
comp osed of all elements which commute with all other elements of G
a
Theorem If jGj p where p is prime and a then G is not simple
a
Pro of Let jGj p and supp ose that G is simple Since G itself is a p group
by we know that Z G G and since G is simple Z G must b e
G But then G is ab elian and its simplicity implies that jGj p This is a
a
contradiction since jGj p so G is not simple
Theorem The N C Theorem If H 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 al l automorphisms of H In mathematical notation
N H C H M where M Aut H
G G
Corollary A unique Sylow p subgroup is normal
a
Lemma Let jGj p m where a m and p does not divide m If
G is simple then n G satis es al l of the fol lowing
p
n divides m
p
n mod p
p
jGj divides n
p
Corollary Let P be a Sylow p subgroup of G Then n jG N P j
p G
and n divides jG P j
p
Theorem Let H G with jG H j n Then there exists N G such
that N H and jG N j divides n In particular if n and jGj does not
divide n G is not simple
Corollary Let H G and jG H j p where p is the smal lest prime
divisor of jGj Then H G
C and Theorem Let B and C be cyclic of order n Then B
there are exactly n di erent isomorphisms that map B to C
Theorem Every two Sylow p subgroups of G are conjugate
Some History
H older
The range problem was initiated by Otto H older in
Before H older published two pap ers that considerably contributed to the
emphasis on this problem The rst was published in in the Mathema
tische Annalen It was a pap er primarily dealing with the solution of
equations However what was evolving into group theory thanks to Evariste
Galois who we will discuss in chapter seems to have proved useful to his
work The concepts of normal subgroups and a comp osition series are dis
cussed A comp osition series is a series of normal subgroups
G G G G G
0 1 n 1 n
where no normal subgroups exist b etween each G i e each subgroup is max
i
imal normal in the next The factor groups G G are all simple groups
i i 1
What is so imp ortant ab out this series is expressed in the Jordan H older the
orem which states that these simple groups called comp osition factors are
uniquely determined up to isomorphism Thus it is apparent that the comp o
sition series acts as a typ e of ngerprint for a group H older was among the
rst to recognize that the comp osition factors are building blo cks of groups
and deserve sp ecial study It was the second pap er published in in which
H older states
It would b e of the greatest interest if a survey of all simple groups with
a nite numb er of op erations elements could b e known
One other advancement of this time deserves recognition Group
theory was evolving into a sub ject in its own right and the idea of treating
groups in the abstract an idea attributed to Cayley was nally b eing accepted
In his start up on the range problem H older was the rst to study groups in
the abstract More often in the past groups were considered with resp ect to
their mo de of representation for example a linear transformation group The
range problem initiated the typ e of exploration that only required knowledge
of the order of the group
H older studied groups having orders from to He did not dis
cover any new simple groups since the unique simple group of order was
known to b e simple it is A and will b e discussed b elow as was the group of
5
order PSL found by Jordan in His metho ds were imp ortant
2
however since they were used by all others working later on the range prob
lem His ideas provided imp ortant general theorems which can b e and were
used within any range and will b e discussed b elow
The most useful to ols that H older employed were the Sylow theorems
H older was comfortable with p ermutation groups and also used this theory
Many of the lemmas that he used in more general theorems came from p ermu
tation 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 pro duct of
three or fewer primes not necessarily distinct H older proved that groups with
2
orders pq p q or pq r are not simple Sylow had already taken care of those
a
groups with orders p These theorems can b e proven in a more e ective
manner using only the Sylow theorems which Burnside did in later years The
following pro ofs are similar to the metho ds used by Burnside rather than the
p ermutation theory used by H older
H older s Pro ofs Using the Sylow Theorems
Theorem If jGj pq where p and q are primes then G is not simple
Pro of Let jGj 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 n is n q since n must divide q by but cannot equal by
p p p
and our assumption This implies that q mo d p which is a contradiction
since p q Thus our assumption is false and G is not simple
2
Theorem If jGj p q where p and q are primes then G is not simple
2
Pro of Let jGj p q where p and q are primes and assume G is simple The
2
choices for n are n p or p Supp ose that n p Then p mo d q so
q q q
p q But the only choice for n is q which implies q mo d p thus q p
p
2
So n p which means that n p Let us now count elements in the group
q q
2 2
Since n p we have p subgroups each with order q Notice that they have
q
prime order which means that they are cyclic and have no two elements in
2
common except for the identity This means that there are p q elements
with order q Let denote the numb er of the rest of the elements Then
2 2 2 2
jGj p q or p q p q p
Thus there are enough elements not of order q to only t into one Sylow
p subgroup which means there is a unique Sylow p subgroup which must b e
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
unique thus normal Thus our assumption was wrong and G is not
simple
Theorem If jGj pq r where p q and r are primes then G is not
simple
Pro of Assume jGj pq r where p q and r are primes and assume G is
simple Without loss of generality we may assume that p q r The
p ossibilities for the size of S y l G are as follows
p
n q r or q r
p
n p r or pr
q
n p q or pq
r
Notice that we may eliminate q and r as p ossibilities for n since p q r
p
using Sylow s rd Also we may eliminate r as p ossibility for n for
q
the same reasons We may eliminate q as p ossibility for n since we know that
r
jGj cannot divide q since there is no p factor in q We conclude that there
are four cases only
n q r n q r n q r n q r
p p p p
n p n p n pr n pr
q q q q
n p n pq n p n pq
r r r r
If we examine each of these cases by counting elements we nd that none are
feasible
The First Case we can conclude that the numb er of elements with
order p is q r p the numb er of elements with order q is p q and
the numb er of elements with order r is p r Note that this is p ossible
since each Sylow subgroup has prime order so no two Sylow subgroups of the
same order have elements in common except for the identity If we add the
numb er of elements that we have so far it is q r p p q p r
pq r q r pq p pr p or pq r q r pq pr p Note that q r pq is
p ositive since p r and pr p is p ositive if r zero otherwise Thus
we have pq r some p ositive numb er as the numb er of elements in G which is
a contradiction Thus the numb er of Sylow subgroups is not the rst case
The Second Case Using the same arguments as ab ove the second
case provides us with the following numb er of elements q r p p q
pq r but this is equal to pq r q r pq p pq r pq pq r pq r q r p
or pq r q r p p and we see that q r p p must b e p ositive Thus
again we have over pq r numb er of elements which is a contradiction
The Third Case Using the same arguments as ab ove the third case
provides us the following numb er of elements q r p pr q p r
which is equal to pq r q r pq r pr pr p pq r pq r q r 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 numb er of elements q r p pr q pq r which equals
pq r q r pq r pr pq r pq pq r pq r q r pq r p q r and note
that pq r q r is p ositive as is pq r p q r if q r q r which is true if
q and r are and resp ectively which they are Thus we have another
contradiction which implies that our original hyp othesis was incorrect and G
is not a simple group
The p ower of these theorems along with a few others eliminated all
but seven orders out of the rst cases The seven remaining groups had
orders known to b e simple known to b e simple
and H older was able to show that all but and were orders of non
simple groups using various techniques of p ermutation group theory yet it has
b een said that his ability to use p ermutation groups was somewhat lacking It
did take him nearly twenty pages of calculation to demonstrate that groups of
order and were not simple
Cole Burnside
It was an American mathematician who followed the path laid by
H older Frank Nelson Cole continued the range problem in
examining groups with orders ranging from to The metho ds used
by H older were also used by Cole The Sylow theorems provided the most
p owerful to ol of investigation and Cole also lo oked at groups in the abstract
sense only recurring when convenient to their representation in terms of sub
stitutions of n letters p ermutation groups H older s theorems of three
or fewer primes proved useful to eliminate all but groups b etween and
Sylow s theorem that n mo d p eliminated another Eventually
p
3
Cole determined that A PSL and PSL groups of orders
6 2 2
and resp ectively were the only simple groups with order b etween and
The simple group of order was never recognized as simple b efore
Cole s work even though it had b een discussed by mathematicians such as
3
Mathieu and Kirkman It was classi ed later as PSL following the ad
2
vancements made by Dickson and Mo ore It was a sp ecial discovery in more
ways than one since it launched the work of Eliakim Hastings Mo ore
n
who discovered that the in nite family of groups PSL p was simple
2
n
except when p or This in turn led to the pro of by Dickson that the
n
in nite family of groups PSL p are simple which is a generalization of
m
Jordan s original result This family shall b e discussed further in the
subsequent chapter Notice that there were no new metho ds evident in Cole s
work the Sylow theorems served him well
William Burnside who has b een called the rst real
group theorist in history b ecause 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 p ermutation group theory He
did develop some arithmetic tests the most imp ortant of which states that
a simple group of even order must b e divisible by either or The
understanding of p ermutation groups had advanced since H older 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 p ermutation theory
using only the abstract ideas such as conjugacy classes and normalizers Burn
side claimed that even in reference to the pro ofs of the Sylow theorems from
the p oint of view of the right metho d they leave something to b e desired
He subsequently rewrote them Notice that the pro ofs given ab ove of H older s
three or fewer primes theorem are essentially Burnside s rewrites Not only
did Burnside simplify the pro ofs for these but he also extended the theorem
to include combinations of four or fewer primes A couple of his pro ofs are
given b elow
Burnside Theorems of Four or Fewer Primes
3
Theorem If jGj p q where p and q are primes then G is not simple
3
Pro of Assume jGj p q where p and q are primes and assume G is simple
2 3
The choices for n are n p p or p and n q which implies
q q p
that q p Supp ose that n p Then p mo d q which contradicts q p
q
3 3
Supp ose that n p Count elements there are p subgroups each with
q
3
order q which have trivial intersections Thus there are p q elements
with order q Let denote the numb er of the rest of the elements Then
3 3 3 3
jGj p q or p q p q p
Thus there are enough elements not of order q to only t into one
Sylow p subgroup which means there is a unique Sylow p subgroup which
must b e normal in G But this is a contradiction to our assumption that G is
simple which leaves us with the last p ossibility
2 2 2
Supp ose that n p Then p mo d q q j p q j
q
p p Since q is prime this implies that q j p or q j p Since
q p q j p only But this implies that p q p so q p and
p and q are consecutive primes But the only consecutive primes are and
so if G is indeed simple p and q is the only p ossibility Thus if we
3
show that a group of order is not simple we have a contradiction
3
Supp ose jGj Then n or But note that jGj do es
2
not divide Thus n which implies that G is not simple So the only
2
p ossibilities left are that one of the original Sylow subgroups is unique thus
normal contradicting our hyp othesis that G is simple
2 2
Theorem If jGj p q where p and q are primes then G is not
simple
2 2
Pro of Assume jGj p q where p and q are primes and assume G is simple
Without loss of generality we may assume that q p The choices
2
for n are n p and p If n p then p mo d q which contradicts the
q q q
fact that q p
2
So supp ose that n p Then the argument from the ab ove pro of
q
2 2
holds i e p mo d q q j p q j p p Since q is prime
this implies that q j p or q j p Since q p q j p only But
this implies that p q p so q p and we can show that a group of
2 2
order is not simple
2 2
Let jGj Then n or But is not mo d and
3
jGj do es not divide thus the Sylow subgroup is unique thus normal in
G So the only p ossibilities left are that one of the original Sylow subgroups is
unique thus normal contradicting our hyp othesis that G is simple
In Burnside completed the range problem up to order
Shortly after this time b eginning in a new technique emerged develop ed
by Burnside and Georg Frob enius called character theory This
theory which is based on the study of certain functions characters from
a group into the complex numb ers has made a great impact on the study
of simple groups through this century It was character theory that provided
Burnside with a pro of of a monumental theorem that follows and outshines
the four or fewer primes result In Burnside proved that any group with
a b
order p q where p and q are prime is not simple unless it is of prime order
Obviously this theorem plays a signi cant role in the simpli cation of the
work required on the range problem after The extension of this result to
a b c
orders made up of a combination of three primes p q r has b een a di cult
problem which has lasted until the present and the metho d of investigation
has most often b een character theory Unfortunately it is b eyond the scop e
of this pap er to mention character theory in more depth
The Completion of the Range Problem Through Order One
Million
The turn of the century saw two mathematicians George Abram
Miller and his student G H Ling work on the range problem for
orders b etween and in The original techniques of investigation
had not changed much however there were a couple of new results which came
a a 2 a b
from the older metho ds One was that any group of order p q p q and p q
for a p q was not simple Notice that these results were the
previews of what was to come in Burnside s theorem The problem of
o dd 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 go o d
use was the fact that there were no simple groups with o dd orders less than
There was increased work on the theory of p ermutation groups and on
transitive groups in particular which help ed with the investigation of individual
orders A p ermutation 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
b etween and There seemed to b e quite a gap after the work of Miller
in interest in the range problem It was not until that anyone approached
the orders following This may have b een due to the di culty that the
larger orders presented and the lack of new results which would act quickly
and sweepingly although one must rememb er the Burnside theorem which did
exactly that It was not until that new metho ds actually arose to handle
groups of particular orders
While work on the in nite 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 sp oradic through
the twentieth century L P Sicelo was the next mathematician to tackle the
orders through in He found simple groups with orders
and He was not able to prove the uniqueness of the simple group
with order and it was not until that Miller successfully showed
that the group was A and unique Cole came back to the game in with
7
the orders through He found four simple groups having orders
and He found di culty with the uniqueness of two
orders and Both of these are unique simple groups as shown by
Richard Brauer in using character theory It to ok eighteen years to nd
the metho ds to complete this task The next time that someone chipp ed away
at the range problem was in Michaels to ok the task of showing that
the unique simple groups b etween and were of orders
and
In Marshall Hall Jr extended the range problem
to order one million He drew together all of the metho ds used from the
late nineteenth century onwards a great deal of the later metho ds relying
on advanced techniques of character theory His assortment of metho ds also
included some computer work Hall s metho ds were unsuccessful with only
orders It was in that two students Beisiegel and Stingl extended
work on the classi cation of simple groups according to the size of their Sylow
subgroups undertaken by Paul Fong The remaining orders were taken
care of and the range problem to one million was complete
It was not necessarily the p eople 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 Dickson listed a
total of known simple groups many memb ers of in nite families of simple
groups see b elow By only three new groups were added to this list M
Suzuki discovered the simple group with order in as he discovered
n
the in nite family S z Z Janko uncovered the simple group of order
in however this group was not a memb er of an in nite family
that is it is a sp oradic simple group In Hall and Janko discovered
a simple group J with order which was also sp oradic None of
2
these three groups was discovered b ecause of work done on the range problem
Apart from these three by those simple groups with orders less than
one million were generally known to b e simple b efore they were encountered
in the course of the range problem They consist of classical linear groups
alternating groups and the Mathieu groups
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 b elow
Easy violation of Sylow s third theorem and use of Corollary
2
Example If jGj then G is not simple
We only need to lo ok at the p ossible numb er of Sylow subgroups to
show that there is only one thus it must b e normal by Note that by
Lemma the numb er of Sylow p subgroups must divide the remaining
numb ers left in the order of the group Thus we have n or
13
Only mo d thus n
13
2
Example If jGj then G is not simple
This works in the same manner as ab ove we shall lo ok for the numb er of
Sylow subgroups to show that there can only b e one
2 2
n or
11
If we check each none except is mo d If we had lo oked rst at
n we would have found that n could b e which is mo d
13 13
3
Example If jGj then G is not simple
This is an even order that works in the same manner Notice the large
numb er of p ossibilities for n
17
n or
17
However none of these are mo d thus the Sylow subgroup is
solitary and normal
Easy violation of Lemma
2 2
Example If jGj then G is not simple
The p ossibilities for n are and
11
If we ignore for the moment we can exclude all p ossibilities except
by So if we assume G is simple then n But notice that
11
jGj do es not divide since there is no second factor of in Thus
by we have a contradiction and G is not simple
2 2
Example If jGj then G is not simple
The p ossibilities for n excluding the smallest factors since they cannot
11
b e mo d are and All
except and violate Thus if we assume G is simple then
n Once again jGj do es not divide since there is no second
11
factor of in Thus by we have a contradiction and G is not
simple
Notice that for to work n must b e fairly small Here are a
p
couple of examples where n is to o large to use and a di erent technique
p
is needed counting elements
3
Example If jGj then G is not simple
Assume that G is simple The p ossibilities for n are the following
11
n Notice that only mo d so we may
11
rule out the other p ossibilities Can we also rule out using No
is large enough that jGj j Let us check n for an easier approach
7
n The only p ossibility that do es not violate
7
is and similarly jGj j since is large enough Thus we have
n and n A new strategy is needed for this problem We
11 7
know jS y l G jand jS y l G j and we know that each Sylow and Sylow
11 7
subgroups have and elements in them resp ectively Any group of
prime order is also cyclic and we know that two di erent cyclic groups of
the same order that have more than one element in common must b e equal
Thus each of the elements of S y l G and S y l G must intersect only
11 7
trivially We could count the elements in each We have groups with
distinct elements in each The numb er of elements in S y l G is
11
then and similarly the numb er of distinct elements in S y l G
7
is We have accounted for
elements so far There are only elements in the group so we have a
contradiction Thus our assumption was incorrect and G is not simple
3
Example If jGj then G is not simple
This is similar to the ab ove order Assume that G is simple Note the
p ossibilities n and n Us
13 5
ing we nd that n and n and b oth numb ers are
13 5
to o large to use Noticing that the subgroups in S y l G and
13
S y l G are of prime order thus cyclic we may count elements We
5
have Thus we have a
contradiction and G is not simple
The following two are more di cult cases using and
5 2
Example If jGj then G is not simple
Assume that G is simple Notice that n and only
5
mo d Also jGj j So n Notice also that is not
5
2
mo d so we may use the later half which states that there
2
exists S and T S y l G such that S T and jS S T j by
5
contrap ositive This implies that jS S T j This is b ecause S T
2
is a subgroup of S and jS j thus if S T f g which is necessary
2 2 2
if jS S T j then jS T j must b e or It cannot b e b ecause
that would imply that S T S T By Lagrange s theorem we
2
have that jS j jS S T j jS T j Thus jS T j and we
may use Corollary which states that since is the smallest prime
divisor of jS j S T S and by the same argument S T T Consider
the normalizer of S T in G N S T By the previous discovery we
G
have that S N S T and also T N S T Thus S and T must
G G
b e subgroups in the S y l subgroup of N S T Since N S T G
5 G G
by Lagrange s theorem again
2
jN S T j jS j jN S T S j jN S T S j
G G G
2 3 4 5
Thus jN S T S j has to b e or If we lo ok at the
G
numb er of Sylow subgroups in N S T we see that it must also b e
G
2 3 4 5
or dep ending on jN S T S j One further condition
G
4
that n N S T is mo d leaves us with n N S T This
5 G 5 G
4 2 4 2 5
implies that divides jN S T j Thus jN S T j or
G G
2 5
If jN S T j then N S T G and S T G which is a
G G
contradiction to our assumption that G is simple Thus jN S T j
G
2 4
But this implies by Lagrange that jG N S T j Note
G
that jGj do es not divide or alternately any subgroup with index
is normal Thus we have by theorem that G is not simple
a contradiction to our assumption but the last alternative Thus our
assumption was incorrect and G is not simple
5 3
Example If jGj then G is not simple
Assume that G is simple The p ossibilities for n are the following
3
n or
3
eliminates all but and Using and noting that jGj cannot
2
divide we are left with n But is not mo d so we can
3
conclude by that there exists S and T S y l G such that S T
3
2
and jS S T j Using the same pro cess as ab ove we can conclude
2
that jS S T j and by Lagrange jS T j By since is the
smallest prime divisor of jS j S T S and similarly S T T Thus S
N S T and T N S T We have by Lagrange that jN S T j
G G G
3
jS j jN S T S j jN S T S j And since N S T G
G G G
2 3 4 5
jN S T S j or We know that n N S T must
G 3 G
divide jN S T S j and also that n N S T mo d thus
G 3 G
2 4 4
n N S T or If n N S T then jN S T S j
3 G 3 G G
4 5 3 4 3 5
or and jN S T j or jN S T j cannot b e
G G
3 5
since that would make G N S T and thus not simple
G
3 4
Supp ose jN S T j Then jG N S T j and since jGj
G G
do es not divide G is not simple by This is a contradiction
3 4
to our assumption thus jN S T j If n N S T
G 3 G
2 3 2 3 3 3 4 3 5
then jN S T j or We know that
G
3 4 3 5 3 3
jN S T j or Thus supp ose jN S T j
G G
2
Then jG N S T j and jGj do es not divide showing that
G
3 2
G cannot b e simple Supp ose that jN S T j Then
G
3
jG N S T j and still the index is to o small and jGj j Thus
G
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
3 2
Example If jGj then G is not simple
Assume that G is simple The p ossibilities for n are the following
11
n or Only mo d thus n
11 11
Lo ok at one subgroup in S y l G say S S y l G Let N b e the
11 11
normalizer in G of S N N S Then since n jG N j
G 11
we know that jN j by Lagrange Let C b e the centralizer of S in
G C C S We know by the N C theorem that the factor group N C
G
is isomorphic to a subgroup of Aut S The set of automorphisms of S has
order since S is cyclic This implies that jN C j divides
By Lagrange again since jN j jC j jN C j the only choice
for jN C j is thus jC j We see that the centralizer in G of S has
Sylow subgroups Let P S y l C Then jP j Consider N P and
3 G
note that N P cannot equal G since we are assuming G is simple and
G
P N P Clearly P C C S Since P commutes with all elements
G G
of S then S C P But C P N P so S N P which means
G G G G
that jN P j is divisible by By there exists a Q S y l G such
G 3
2
that P Q But jQj so jQ P j By then P Q Thus
2
Q N P which implies that jN P j is also divisible by So the
G G
2 3
least order of N P is which means that jG N P j But
G G
3 3
itself is to o small since jGj cannot divide This implies that G is
not a simple group 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 nd a sub
group of G which has order large enough to make the index of it in G to o
small to b e divisible by the order of G thus utilizing the theorem The
way to nd 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 digging a few layers deep into the
structure of the group
2
Example If jGj then G is not simple
Assume that G is simple The following lists the p ossibilities for all
S y l G subgroups
p
n
11
n
5
n
7
n
3
The numb ers in b old are those that do not violate either or
These numb ers indicate that the only p ossibilities for jN s j where
G p
s S y l by are the following
p p
jN s j
G 11
jN s j or
G 5
jN s j or
G 7
2 2
jN s j or
G 3
Working systematically we shall try to eliminate each of these as p ossi
bilities Supp ose that jN s j Lo ok at jS y l j in N s
G 5 11 G 5
denoted n N s n N s or Note that the only
11 G 5 11 G 5
choice that do es not violate is n N s Thus by
11 G 5
N s N s N s and thus jN s j
G 5 N (s ) 11 G 5 G 5
5
G
N s N s by Lagrange But N s N s is the nor
N (s ) 11 G 5 N (s ) 11 G 5
5 5
G G
malizer in N s of a Sylow subgroup and note that N s is the
G 5 G 11
group of all elements in G that normalize a Sylow subgroup Thus
N s N s N s which implies that divides jN s j
N (s ) 11 G 5 G 11 G 11
5
G
But we know that jN s j from ab ove thus we have a contra
G 11
diction We now know that jN s j and n Supp ose that
G 5 5
jN s j Note that n N s or By
G 7 11 G 7
n N s is the only p ossibility Then by and Lagrange
11 G 7
N s N s But this implies that j jN s j
11 G 7 G 11
N (s )
7
G
which is a contradiction Thus jN s j and n Now lo ok
G 7 7
at the p ossibilities for n N s or By n N s and
5 G 7 5 G 7
by the same argument as ab ove this implies that j N s We have
G 5
from ab ove that jN s j thus we have a contradiction The only
G 5
p ossibility 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 ab out the size
of S y l G more than once to draw a contradiction The following example
p
starts in this manner then requires a metho d previously seen and comes to a
conclusion with the same metho d used at rst
3
Example If jGj then G is not simple
Assume that G is simple The following list the p ossibilities for the sizes
of all S y l G
p
n
2
n
5
n
19
The numb ers in b old indicate those that do not violate or The
following are the p ossible orders of the normalizers of the Sylow subgroups
by
3 3
jN s j or
G 2
jN s j
G 5
jN s j
G 19
3
We would like to determine jN s j so supp ose jN s j Then
G 2 G 2
n N s or We conclude by that n N s
5 G 2 5 G 2
Thus using the same pro cess as ab ove by and Lagrange we can
3
conclude that N s N s Since N s N s
N (s ) 5 G 2 N (s ) 5 G 2
2 2
G G
3
N s then divides jN s j a contradiction Thus jN s j
G 5 G 5 G 2
3 2
and n Note that mo d so by there exist S and
2
2
T S y l G such that S T and jS S T j This implies that
2
jS S T j By we have that S T S and by similar argument
3
S T T Thus S N S T and T N S T so j jN S T j
G G G
3 3 3
In fact our p ossibilities for jN S T j are or
G
3
We may rule out jN S T j since that would imply that
G
N S T G and thus S T G which is a contradiction Supp ose
G
3
that jN S T j Then jG N S T j but jGj j which
G G
3
implies a contradiction by Thus jN S T j Lo ok at the
G
size of S y l N S T n or By n By
5 G 5 5
3
But this N s N S T and Lagrange we have that
5 G
N (S \T )
G
3
implies that j jN s j since N s N S T N s and
G 5 5 G G 5
N (S \T )
G
this is a contradiction since jN s j Thus our original assumption
G 5
must b e incorrect and G is not simple
The orders used for these examples are obviously fairly small As
one can guess the larger the order the more cumb ersome are the choices for
such numb ers as n G Take the simple group J for example This group
p 1
3
describ ed further b elow has order In order to determine
n one must consider p ossiblilities Out of this there are four numb ers
19
which cannot b e eliminated using or Since is the largest prime
divisor of jGj n should b e the most accessible of all sizes of the S y l G
19 p
to nd Imagine what the others must b e like The shear magnitude of the
problems increase as the orders b ecome very large Not all groups of large
order are di cult to handle however Take for example jGj
It is a simple matter of using on the p ossibilities for n that proves G
5
is not simple Nonetheless when the larger orders are di cult they can b e
very di cult They are generally more cumb ersome 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
The Simple Groups
In nite Families of Simple Groups
The Alternating Groups
I have often in my life ventured to advance prop ositions of which I was
uncertain it is to o much to my interest not to deceive myself that I
have b een susp ect of announcing theorems of which I had not the complete
determination subsequently there will b e I hop e some p eople who will
nd it to their pro t to decipher all this mess Galois
The history of group theory itself b egins with the discovery of the rst
comp ositely ordered simple group A The pro cess that led to the discovery of
5
this simple group actually led to the idea of the study of group theory It b egan
with Evariste Galois who led a very short but mathematically
pro ductive life although it to ok time and scrutiny for anyone to understand
his ideas The ab ove quotation was on the nal page written by Galois b efore
he died for so trivial a thing in a duel when he was twenty one years
of age Many of the terms that he used were not rigorously de ned and
his results were not often proven b eing hurriedly jotted on a piece of pap er
Yet Galois did have the rst concept of groups as we de ne them to day and
used them somewhat abstractly in his studies of solvable p olynomials Galois
was working on the p opular algebra problem of the eighteenth and into the
nineteenth centuries the factorability of p olynomials over a eld F
Galois approach to this problem is ro oted in the workings of p ermu
tations The p ossible ro ots of a p olynomial of degree n can b e p ermuted in
n di erent ways For example lo ok at the fourth order p olynomial in the
2 2
complex eld f x x x The four ro ots of the p olynomial are
p p
and Supp ose we let x i i
i
i
p
p
C B
C B
Then we have p ermutations of these four letters such as R
1
A
which switches and and leaves the other two xed There are
similar p ermutations A subgroup of the group of p ermutations can b e
formed in the following way Lo ok at any p olynomial equations involving
or Some equations express a true statement if the numerical values of
2
or are substituted and some do not For example the equation
p
as is for the given values of and An equation is true for
such as is obviously not true The group of p ermutations which
preserve the truth of the true equations form a subgroup of the p ermutation
group Notice that any true equation remains true if and are interchanged
and similarly if and are interchanged Galois called this subgroup of
p ermutations the group of the equation G In our example this group consists
of
C C B C B C B B
C C B C B C B B
I R R R
3 2 1
A A A A
The rst concept of a normal subgoup was b orn by examining the
group G Cho ose a p olynomial expression which is rational in the ro ots
of our original equation but has the following prop erty it s numerical value
t stays xed for some elements of G but changes for others Then those
elements of G which x t form a subgroup H of G Galois showed that if t is
p
a ro ot of the irreducible over F binomial equation x c where p is prime
then the subgroup H is in fact normal in G This pro cess continues to reveal a
metho d of solving equations by radicals and also the inspiration for studying
simple groups Form a new eld F t which is the smallest eld containing
b oth F and t The subgroup H is then the group of the equation over the
new eld F t Rep eat the ab ove pro cess on H to nd a normal subgroup
of H and a new eld F t t where t is the numerical value of the chosen
1 1
expression The pro cess can b e rep eated until we are left with the identity
p ermutation as the subgroup In this case the original equation is said to
b e solvable by radicals over the created eld F t t t Furthermore we
1 n
have a series of normal subgroups much like
H H H H G
0 1 n 1 n
where the index of one in the other H H was shown to b e the prime
i i 1
p
numb er p in the appropriate equation x c This lo oks remarkably like the
comp osition series discussed earlier and since each index is prime we see that
each comp osition factor jH H j must b e trivially simple Galois discovered
i i 1
that an equation was solvable if each index in the comp osition series was prime
and not solvable if some index was not prime This is precisely what happ ens
to quintic equations Some comp osition factor in the comp osition series is
comp ositely simple not having prime order and the end result of the identity
p ermutation 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 By Galois
recognized this group as simple stating The smallest group of p ermutations
which an indecomp osable group can have when this numb er is not prime
is Galois stated this without pro of and it wasn t until
that Jordan would verify this result In fact Jordan gave b etter de nition to
the notion of a comp osition series which was only one great feat of his
work Trait e des substitutions et des equations algebriques 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
purp ose of the Trait e The use of p ermutation groups was still b eing explored
and expanded and groups were generally represented as such Thus Jordan
discovered that the simple group of order which was tied to the quintic
equation was actually the alternating group on ve letters A An alternating
5
group is the subgroup of the p ermutation group made up of all even p ermu
tations A p ermutation is even if it can b e written as a pro duct of an even
numb er of cycles or transp ositions Jordan went further than proving the
simplicity of A He presented a awed pro of for the simplicity of all alter
5
nating groups A for n This was the rst in nite family of simple groups
n
to b e discovered As an example of the p ermutation group theory used the
following is a pro of for the simplicity of A
5
Theorem A is simple
5
5 2
Pro of The cycle structures of the elements in A are the following
5
2
and This notation indicates that there are p ermutations which x
ve letters the identity x one letter and has two cycles x letters and
has one cycle and which has one cycle The orders of the elements in A
5
which are made up of these cycle structures can b e obtained by nding the
least common multiple of the sizes of cycles for each typ e That is the order
of the elements that are made up of two cycles and x one p oint is LC M
of and etc as shown b elow
cycle structure order of elements numb er of elements
5
2
2
The last column ab ove shows the numb er of elements of each order
These numb ers are easily obtained by lo oking at the order of A For example
5
2
2 35
the numb er of elements of order is the numb er of elements of order
3
2
2 35
is etc To show that A is simple we shall pro ceed by contradiction
5
5
Supp ose that A contains a normal subgroup S which is not the identity
5
2
or A itself The p ossible orders of S must divide Supp ose that
5
j jS j Then S contains a Sylow subgroup of A and since S is normal and
5
every two Sylow p subgroups are conjugate S must contain all Sylow
subgroups Thus S contains all elements of order There are elements
of order so jS j accounting for the identity Also j jS j and jS j j jA j
5
so jS j
Now supp ose that j jS j By the same argument as ab ove S contains
all Sylow subgroups and thus all elements of order So jS j thus
jS j Since is divisible by b oth and S must contain all elements of
b oth orders but this is imp ossible if jS j
So supp ose jS j Then S would b e a normal Sylow subgroup
and thus would b e the unique Sylow subgroup But there are elements of
order so this is also imp ossible
Finally supp ose jS j Then jAut S j since
Using the N C theorem N S C S thus N S C S
A A A A
5 5 5 5
Since S is normal in A N S A which implies that C S A This
5 A 5 A 5
5 5
is not true since a counterexample can b e found easily as a cycle which do es
not commute with a pro duct of two cycles So none of the p ossibilities work
and our assumption must b e incorrect Therefore A is simple
5
Simple Groups of Lie Typ e
The remainder of the in nite families of simple groups can b e clas
si ed as Lie groups These include the classical groups the groups of typ e
G the Chevalley groups of typ es E E E and E the twisted groups of
2 4 6 7 8
typ es E and D the Suzuki groups and the Ree groups of typ es G and
6 4 2
F These groups arise as automorphism groups of corresp onding simple Lie
4
algebras In general since the theory of Lie algebras is to o extensive for this
pap er a Lie algebra is a vector space over a eld with a pro duct X Y that
is linear in b oth variables which also meets the following criteria
X X for all X in the vector space
X Y Z Y X X Z X Y the Jacobi identity
The Classical Linear Groups
It was Jordan again in his Trait e who found the next four in nite fam
ilies of simple groups although he was not completely aware of the simplicity
of each Jordan obtained orders generators and the factors of comp osition of
some of these groups and was not explicit ab out the in nite families involved
n
We have seen how the simplicity of the in nite family PSL m p was nally
proven by Dickson in In fact Dickson worked on extending Jordan s
results on all of the linear groups from to Dickson and Dieudonn e
are also credited with further investigating all of the linear groups in the years
to The groups are now known as the pro jective sp ecial 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 rst two are given b elow 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 GL q is the
n
group of all nonsingular linear op erators of a vector space V where V has
dimension n over the eld of order q Thus GL q is a group of n by n
n
matrices The order of GL q can b e given by the following
n
n(n 1) 2 2 n
jGL q j q q q q
n
The subgroup of matrices with determinant is normal and called the sp ecial
n(n 1)
2 n
2
linear group SL q The order of SL q is given by q q q
n n
The center Z of GL q consists of transformations of the form T x x
n
for not The center of SL q can b e denoted Z SL q and the factor
n n
group is the pro jective sp ecial linear group PSL q Its order is given by
n
1
n(n 1) 2 2 n
the following jPSL q j q q q Let the eld b e
n
(n q 1)
the Galois eld GF q where q is a p ower of a prime This group is simple for
n except for PSL and PSL
2 2
Let us lo ok at a sp eci c example of a pro jective sp ecial linear group
The simple group PSL is isomorphic to PSL b oth with order
3 2
We would construct PSL by lo oking rst at GL which consists of
3 3
all nonsingular by matrices over the Galois eld GF For example
is an element in GL The order of GL is the matrix
3 3
3 3 2
Note that this is the same as the order
of PSL and indeed they are isomorphic The reason for this is that all
3
matrices in GL have determinant equal to mo d thus all elements in
3
GL are also in SL The center of SL consists of the identity only
3 3 3
thus SL Z SL SL PSL
3 3 3 3
The construction of PSL isomorphic to PSL b egins with
2 3
GL the group of nonsingular by matrices over GF GL has
2 2
2
order An element in this group lo oks something like
where the matrix entries are mo dulo If we restrict or
ourselves to all matrices in GL with determinant mo d for example
2
we have SL with order The center of GL consists of
2 2
The center of SL are those matrices like
2
matrices in the center with determinant mo d which are only
The simple group PSL is the factor group of SL and and
2 2
these two elements Comparing the order with the formula given ab ove we
1
2
see that jPSL j
2
2
Projective Symplectic Supp ose that the vector space V from ab ove
has a skew symmetric bilinear non singular scalar pro duct so that x y
y x and x x The symplectic group S p q where n m consists
n
of those linear transformations which preserve the ab ove symplectic form In
particular if A B C and D are m m matrices then the transformation
A B
is symplectic exactly when the following represented by the matrix
C D
t t t t t t
hold A C C A A D C B I and B D D B The
pro jective symplectic group P S p q is the factor group S p q Z S p q
n n n
where Z S p q the center of S p q is made up of scalar matrices P S p q
n n n
is simple except for P S p P S p and P S p The order of P S p q is
2 2 4 n
2
m 2m 2m 2 2
given by the following formula q q q q q
An example of a pro jective symplectic group is P S p which con
2
tains elements and is isomorphic to A S p is a subgroup of GL
6 2 2
the set of matrices over the Galois eld of elements First we construct
2
GF by lo oking at the irreducible p olynomial x over Z We nd that
3
2
GF fax b hx ig and the elements are
f x x x x x x g
Following the equations ab ove and simplifying our example by only lo oking
at elements for which B C we can write a couple of elements of S p
2
x x
The two elements in Z S p are and
2
x x
and since P S p is the factor group of S p and these two and
2 2
x x
are also in P S p and elements we know that
2
x x
One can easily verify that x x and x x in GF
Other Lie Groups
A brief mention of the history of other groups of Lie typ e is in order
During the p erio d to a new family of simple groups of Lie typ e was
discovered by Dickson Until classical linear and this new family were
the only simple groups of Lie typ e known Claude Chevalley pro duced a much
needed new way of approaching these simple groups and in the pro cess he
discovered several more in nite families of simple groups of Lie typ e These
are referred to as the Chevalley groups Chevalley s progress on the groups of
Lie typ e successfully increased the interest in the eld and it wasn t long b efore
new in nite families of simple groups of Lie typ e were found In particular
in Suzuki discovered his in nite family while working on what is now
called a classi cation problem see Chapter He was trying to nd all
simple groups in which the centralizer of an involution that is all elements
n
that commute with a particular element of order two is a group of order
n
In the pro cess of trying to eliminate all p ossibilities except for PSL
n
and PSL n Suzuki found another family with the given prop erty
n
These are S z In Rhimak Ree was analyzing the Suzuki groups using
a particular metho d Steinb erg s which had pro duced in nite families of Lie
typ e b efore and came up with two additional families Thus the Chevalley
Steinb erg Suzuki and Ree groups are the simple groups of Lie typ e along
with the classical linear groups
The Sp oradic Simple Groups
The remaining known simple groups do not t into any large mo del
of similar attributes as do the in nite families They were discovered often
one by one Some do t together by metho d of discovery or by construction
We will examine these prop erties brie y b elow First the following table lists
the sp oradic simple groups their order if not to o large their discoverer
according to some references and the date of their discovery
Name Order Discovered by Date
4 2
M Mathieu
11
6 3
M Mathieu
12
7 2
M Mathieu
22
7 2
M Mathieu
23
10 3
M Mathieu
24
3
J Janko
1
7 3 2
J H aJ Hall Janko
2
7 5
J HJM Janko Higman McKay
3
9 2 3
HS Higman Sims
7 6 3
M cL McLaughlin
13 7 2
S uz Suzuki
10 3 2 3
H e Held Higman McKay
21 9 4 2
C o Conway Leech
1
18 6 3
C o Conway
2
10 7 3
C o Conway
3
17 9 2
F i Fischer
22
18 13 2
F i Fischer
23
0
F i Fischer
24
Ly Lyons Sims
14 3 3
R u Rudvalis Conway Wales
0 9 4 3
O N O Nan Sims
M Fischer
B Fischer
15 10 3 2
F Thompson Smith
3
14 6 6
F Fischer Smith Harada
5
J Janko
4
Table The Sp oradic Simple Groups
The names of discoverers followed by a star are those which have
some discrepancy dep ending on sources Those orders denoted by a star are
to o large to t this table For example the order of the group M the largest
of the sp oradic simple groups is
The Mathieu Groups
The Mathieu groups M M M M and M are the earliest
11 12 22 23 24
sp oradic simple groups to b e discovered They were describ ed by Emile Math
ieu in and Mathieu was in uenced by Cauchy s work
on p ermutations Mathieu was investigating multiply transitive functions and
thus p ermutation groups and multiply transitive p ermutation groups A p er
mutation group on a set A is said to b e n transitive if for any ordered pair of
n tuples of elements of A there exits some element of the p ermutation group
that maps one tuple to the other That is x g y for i n where x
i i i
and y A and g p ermutation group of A A transitive function is one
i
which is left invariant under the p ermutations of a transitive group which
was discovered by Mathieu In the course of his work Mathieu attempted to
extend transitivity by constructing an n transitive p ermutation group out of
a n transitive p ermutation group He was able to nd an algorithm for the
construction of these groups when their construction was p ossible The high
est transitivity found in a simple group is transitive and Mathieu discovered
the transitive p ermutation groups on symb ols and on symb ols which
are M and M The other Mathieu groups arose as subgroups of these and
12 24
a subgroup of M For example M is the subgroup of M formed as the
23 11 12
stabalizer of a p oint in M Each of the Mathieu groups are multiply transi
12
tive The simplicity and uniqueness of the Mathieu groups was not expressed
until the s in a pap er by Witt who was describing what is called the
Steiner system The Mathieu groups are now normally describ ed in terms of
this system
Centralizer of Involution Problems
The next sp oradic simple groups were not discovered until around
one hundred years after Mathieu s nd The rst of these is Janko s rst J
1
and the metho d by which it was discovered b ecame an imp ortant part of the
theory of simple groups and an imp ortant metho d to discover other simple
groups The central feature of the metho d 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 in nite families of simple groups of
Lie typ e The centralizer of an involution as an entity is imp ortant due to a
numb er of results Two of these are a theorem due to Brauer and Fowler and
the Feit Thompson Theorem or Odd Order theorem which are b oth lo oked
at b elow
Because an involution is an element of order two the order of a group
containing an involution must b e divisible by two If it were guaranteed that
a simple group contained an involution this may increase the p otential of
classifying simple groups according to something related to involutions This
result was indeed obtained in the Feit Thompson Theorem It was not a swift
theorem to come up with however and the o dd versus even order of simple
groups was a long standing question In fact conjectures on this question date
back to and Burnside Burnside had a go o d hunch that simple groups
must necessarily have even order and from to he attempted to show
this He was successful at proving that all simple groups with orders under
had even orders yet he could not generalize his result He b elieved that
the necessary technique to prove his conjecture was character theory The
problem came alive again in 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 ab elian has
even order This result was extended in by Feit Hall and Thompson
who proved that a simple group must have even order if the centralizer of any
non identity element is nilp otent i e all of its Sylow subgroups are normal
Three years later the same two Feit and Thompson to ok pages of the
Paci c Journal of Mathematics to prove that all groups of o dd order are
solvable This means that the comp osition series of a group of o dd order
contains comp osition 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 p ertaining directly to the centralizers of involutions was
actually found earlier than the Feit Thompson Theorem In Brauer
and Fowler proved that there are at most a nite numb er of simple groups in
which the centralizer of an involution has a given structure
Theorem If G is a nite simple group of even order and t is an involution
2
in G then jGj jC t j
G
C t denotes the centralizer in G of t Since there can only b e a nite
G
numb er of groups with orders less than a particular numb er then there are
only a nite numb er of groups with the centralizer of an involution isomorphic
to a given centralizer This provided the idea of at least some classi cation
of nite simple groups by the structure of the centralizer of involutions The
imp ortance of this result was furthered by the Feit Thompson Theorem since
then the result p ertained to all simple groups not just simple groups of even
order This theorem has b een improved up on 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 subgroup In general it has b een 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 p owerful results which may allow for
the characterization of a simple group by its centralizer of a central involution
An example of this typ e of theorem is the following due to Brauer
Theorem Let G be a simple group which contains an involution whose
centralizer is isomorphic to GL q factored by a subgroup of odd order in the
2
center of GL q and where q is an odd prime power congruent to mod
2
Then either
PSL q or G
3
M and q G
11
There are many other such theorems and the theory involved in the study of
centralizers of involutions is extensive This pap er will only b e able to concern
itself with a brief description of the discovery of some of the sp oradic simple
groups due to centralizer of involution theory
Let us return to the next sp oradic simple group to b e discovered J
1
The story of Janko s rst group b egins with the centralizer of the involutions
n
in one family of Ree groups of Lie typ e denoted R It was found that the
n n
centralizer of an involution of R is isomorphic to the group Z PSL
2 2
the external direct pro duct It was also noted that the Sylow subgroups are
elementary ab elian of order Thus an interesting task b ecame to determine
all simple groups with Sylow subgroups with the ab ove prop erties which have
n
centralizers of involutions isomorphic to Z PSL p p an o dd prime For
2 2
n
p the new simple group J was discovered Janko proved the following
1
theorem
Theorem If G is a simple group with ab elian Sylow subgroups of order
and the centralizer of an involution of G is isomorphic to Z PSL
2 2
then G is a uniquely determined simple group of order Moreover
G is isomorphic to the subgroup of GL generated by the following two
7
elements of order and
and S S
2 1
Janko was the lucky receptor of further inspiration which led to two
other sp oradic simple groups J and J After the discovery of J Janko
2 3 1
lo oked further for p ossible 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 by A He actually found two new
5
groups with the same centralizer of an involution J and J Hall and Wales
2 3
proved the existence of J and Higman and McKay proved the existence of
2
J
3
The question of the existence of two simple groups with isomorphic
centralizers of involutions led to the discovery of the next sp oradic simple
group in the story We now cease chronological order D Held knew that
the groups M and PSL have involutions with isomorphic centralizers
24 5
While investigating this phenomena Held discovered yet another simple group
with the same centralizer of an involution H e This is the only case of three
simple groups with isomorphic centralizers of involutions
The next sp oradic simple group to b e obtained by examining cen
tralizers of involutions is Ly John McLaughlin s group M c to b e discussed
b elow has a centralizer of an involution which is isomorphic to the group A
8
which denotes the p erfect extension of A by Z The idea then arose to study
8 2
centralizers of involutions which are isomorphic to A for n On such
n
an investigation Richard Lyons who was a student of Thompson s made the
following discovery
Theorem If G is a simple group in which the centralizer of an involution
is isomorphic to A n or then n and
n
8 7 6
jGj
In fact the result was shown that simple groups could only arise from central
izers of involutions isomorphic to A and A Incidentally It was Janko who
8 11
had worked on this problem He showed that when n and there were
no simple groups with the said centralizer of an involution He gave up b efore
working on n He did however discover the last sp oradic simple group
falling under the category of centralizer of involution problems and that was
J
4
Rank Permutation Groups
The group J has a structure which b ecame imp ortant to the con
2
struction of four more sp oradic simple groups J is said to b e a primitive rank
2
p ermutation group A group G has p ermutation rank r if G is transitive on
a set and the subgroup of G that xes a p oint of has exactly r orbits on
Recall that a group is transitive if for a set and any two elements and
there exists an element g G such that g Also the subgroup
of G that xes a p oint of are those elements in G for which g The
orbits on are sets of the form f g j g Gg The orbits of partition
The group J ts this description if one considers the maximal subgroup of
2
index H J The p ermutation representation of J on the right cosets
2 2
of H is a transitive action which pro duces a primitive p ermutation represen
tation of J of degree i e J is a transitive p ermutation group of degree
2 2
which takes the role of the set ab ove On this set H xes one p oint
and its action pro duces two other orbits rendering J a rank p ermutation
2
group In fact the existence of J was proven using the theory of rank p er
2
mutation groups The maximal subgroup H happ ened to b e isomorphic to the
simple pro jective sp ecial unitary group U Donald Higman and C Sims
3
noted the similarity in p ermutation prop erties b etween the groups U and
3
M and in record time were able to construct a new simple primitive rank
22
p ermutation group using M as the maximal subgroup and extending it
22
to obtain the group HS A similar technique was used by McLaughlin who
started with the group U to extend it to a rank p ermutation group that
4
is simple called M cL Suzuki obtained his sp oradic simple group S u in the
same manner starting with G a Chevalley Lie typ e simple group Finally
2
the fourth rank p ermutation group was constructed by Rudvalis using the
2
Ree group F It is R u
4
The Remaining Sp oradic Simple Groups
This shall serve to brie y describ e the discovery of the remaining
sp oradic simple groups The Conway groups C o C o and C o came out
1 2 3
of the study of an automorphism group of a lattice called the Leech lattice
which is determined by a set of vectors in dimensional Euclidean space
with integral co ordinates The three simple groups happ en to have b een sub
0
groups of this automorphism group The Fischer groups F i F i and F i
22 23
24
were discovered by Fischer while studying classes of transp ositions These
are conjugacy classes generated by involutions such that the pro duct of two
involutions in the class has order or Fischer generated groups by these
classes and put further conditions on the groups proving that the new group
is either a symmetric group a certain classical group or one of the three Fis
cher groups Fischer then turned to groups generated by f g transp ositions
two involutions in a class have a pro duct of order or Two groups
B and M or Baby Monster and Monster were discovered The Monster is
the largest sp oradic simple group and a representation for it was obtained by
hand by Rob ert Griess It was in terms of square matrices that were
by in size The groups B F and F are actually subgroups of the
3 5
Monster F was found by Thompson and F is attributed to Harada Norton
3 5
and Smith The O Nan group came out of the study of groups with particular
Sylow subgroup structure
The metho ds used to discover new sp oradic simple groups were often
haphazard as Daniel Gorenstein says some of the groups seemed literally
plucked from thin air Sometimes the techniques used were character
theory based 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
describ ed there is the existence and there is the uniqueness Often several
di erent 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 J had the timely advantage
1
of computers to aid in their discovery existence and uniqueness questions
The Classi cation Theorem
History
The study of centralizers of involutions proved not only very useful
in lo cating certain sp oradic groups but also marks what some would consider
the start of the classi cation pro ject As noted ab ove in Brauer made
his great discovery that there are only a nite numb er 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 go o d results were obtained often with the
discovery of sp oradic simple groups Brauer s ideas served to provide a new
avenue down which some could dream of an overall classi cation of all nite
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 ab out Lie groups in the
s is another But in the s there was still much to b e accomplished
b efore a classi cation idea could b ecome a reality
The s provided the study of simple groups with some of those
high p owered results it needed The most in uential is the famous Feit
Thompson theorem or the o dd order theorem which states that all groups of
o dd order are solvable It was not only the result that was terribly in uential
but also the structure of this page pro of Thompson was also resp on
sible for another very imp ortant result which to ok pages and six years
to complete This is the classi cation 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 subgroup in
Sylow subgroups were b ecoming as telling as centralizers of involutions and
many results stemmed from their study In particular Gorenstein and Walter
characterized simple groups with dihedral Sylow subgroups also in In
Walters classi ed simple groups with ab elian Sylow subgroups These
are general characterizations An example of a sp eci c characterization is
Glaub erman s Z theorem of which showed that every involution is con
jugate to another involution in its centralizer These are only a few of the
imp ortant steps taken in the s and many other results were to follow
By the s there were many roads to classi cation although no
systematic idea of its achievement There were also many sp oradic groups
discovered in the s and some wondered if there was an endless supply of
them Thus in at the University of Chicago when Gorenstein presented
his idea of a step plan to classify all simple groups not many were opti
mistic Gorenstein pro jected that to complete his program would take ab out
thirty years The task seemed daunting yet a few tackled p ortions of the
plan The pro ject was prop elled rather suddenly by a newcomer Michael As
chbacher who came on now like a whirlwind moving directly to a leadership
p osition and sweeping aside all obstacles as he proved one astonishing result
after another The results b eing made at this time were obviously highly
complex and therefore cannot b e handled in this pap er It should b e noted
that the original plan of years was decreased to an actual years and
Gorenstein attributes this to Aschbacher The completion of the classi cation
theorem to ok place in January of
Some Metho ds
The metho ds used in pursuing the idea of the complete classi cation
of nite simple groups naturally changed as progress was made As can b e
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
subgroups that are restricted structurally such as ab elian are go o d candi
dates for the use of character theory in examining them closely However as
the questions ab out the group s internal structure b ecame more broad new
techniques were needed These techniques are called local group theoretic anal
ysis or lo cal analysis It was the Feit Thompson theorem that initiated the
practice of lo cal analysis The predecessors of the Feit Thompson theorem
Suzuki s ab elian centralizer result and the Feit Thompson and Hall result on
nilp otent centralizers see p used character theory to develop the lattice
of prop er subgroups of the group in question This required analysis of ev
ery subgroup This pro cess could not b e used in the Feit Thompson theorem
since there was no information on the structure of centralizers to rely on A
new set of techniques was develop ed by Thompson and their main emphasis
was to lo ok at centralizers and normalizers of prime p ower order subgroups
and analyze their relationships A new term was coined for the normalizer of a
nonidentity prime p ower subgroup and that was lo cal or p lo cal subgroup p
b eing the prime p ower Thus the techniques of lo cal analysis are the metho ds
of examining lo cal subgroups
The lo cal analytic metho ds were explored further by Thompson in his
classi cation of minimal simple groups and his N group theorem of An
N group is a simple group whose lo cal subgroups are each solvable Thompson
explored all p ossible simple groups tting this description and was able to
classify the N groups
Theorem If G is a simple N group then G is isomorphic to one of the
fol lowing groups
PSL q where q
2
2n+1
S z q where q n
2 0
PSL U F A or M
3 3 4 7 11
2 0
U is a unitary group and F is a Ree group of Lie typ e Thompson s
3 4
strategy was to show that an arbitrary N group has internal structure that
lo oks like one of the groups listed Then resemblance was shown to b e actual
isomorphism This pro cess is mirrored in the classi cation theorem as will b e
seen b elow One concept that was invaluable to Thompson s N group theorem
and later to lo cal analysis in general was the idea of emb edded subgroups An
example of a typ e of emb edded subgroup is a strongly emb edded subgroup
M of G This means that jM j is even and the following hold
C t M for every involution t of M
G
N S M for each Sylow subgroup S of M
G
Strongly emb edded subgroups themselves were actually classi ed by Bender
in as either PSL q U q or S z q for q even
2 3
While lo cal analysis was develop ed and results of a di erent na
ture were obtained b ecause of the change of emphasis there were also further
changes in direction by creative individuals A couple of these di erent ap
proaches are mentioned now The metho d of b oth the Feit Thompson theorem
and many classi cation theorems that followed was generally to lo ok 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 pro cedure to achieve this was to examine relatively small subgroups to
develop the lo cal subgroup structure Helmut Bender changed this approach
in his attempt to simplify the pro of of the Feit Thompson 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 ab out
ab elian Sylow subgroups and Gorenstein s and Walter s result ab out dihe
dral Sylow subgroups Originally Bender was lo oking for a revision of the
classi cation as a whole b eginning with the Feit Thompson theorem but his
metho d b ecame a useful to ol in itself
Another innovation that was second only to lo cal analysis techniques
was Fischer s internal geometric analysis We have seen the work of Fis
cher with resp ect to the discovery of sp oradic simple groups His ideas of
transp ositions went much further than only the discovery of his sp oradic
groups however Recall that a class of transp ositions is a conjugacy class of
involutions where the pro duct of any two has order or 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 p ermutes 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 de nitions of connected and nonconnected came from the
nature of the graphs We will see that these play a very imp ortant role in the
classi cation theorem
The Theorem
The entire classi cation theorem is a monumental enterprise of b e
tween and pages taken from around contributors and writ
ten over a p erio d of more than years There are articles stretching out
among p erhaps 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 s and complete classi cation was obtained in the early s We
have seen that a systematic approach to the classi cation was prop osed as
late as The theorem itself states that all nite simple groups have b een
found That is any nite simple group is isomorphic to one of those already
discovered
Theorem Main Classi cation Theorem Every nontrivial nite simple
group is isomorphic to one of the fol lowing
An alternating group
One of the above mentioned sporadic groups see p
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 supp ose that the group G has a set of prop erties X
Given this information one can prove that G is actually a known simple group
deriving a contradiction The inductive nature of the pro of is imp ortant for
lo oking at internal prop erties of subgroups of G For example there is a result
which states that
Theorem Given a minimal simple counterexample G with a set of prop
erties X if H G and K H then H K is a simple group with properties
X
An alternate form of the classi cation theorem which makes the inductive
nature obvious is the following
Theorem Main Classi cation Theorem alternate form If G is a nite
simple group each of whose proper subgroups is a known simple group then G
is a known simple group
Supp ose that we have a minimal counterexample G with X prop erties
that we assume is not a known simple group This assumption forces us
to consider the internal prop erties of G to b e as complicated as any nite
group We cannot assume that G lo oks 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 lo ok like a known simple group Obviously this is
not an easily accomplished task and many of the high p owered lo cal analysis
techniques must b e used carefully in the examination of the internal structure
of the group There are many p ossibilities for a group G with X prop erties
and each must b e considered This accounts for much of the complexity and
length of the theorem since there are around di erent paths to follow
to show that G lo oks like a known simple group The paths themselves are
determined by the prop erties of G so each case is di erent The classi cation
theorem is complete in that it exhausts all of the p ossible structures of G and
leads all p ossible simple groups to the structure of a known simple group In
order for us to know that our simple group lo oks 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 lo oks like a known simple group then the steps toward isomorphism
must b e taken That is internal resemblance must b e shown to b e actual
isomorphism
It is to b e noted that the structure of the classi cation theorem is
very similar to that of the Feit Thompson theorem In fact one can break
down the pro cess of b oth into three steps
Use the given prop erties of G to determine the structure and
emb edding of maximal subgroups containing or intersecting centralizers of
involutions by lo cal analysis
Eliminate as many of these p ossible con gurations by using char
acter theory on smaller groups lo cal analysis on larger groups and arithmetic
metho ds
Use recognition theorems generators and relations to prove that
the only p ossible con guration left is isomorphic to a known simple group
Beginning with the last step rst each of the known simple groups
must b e recognizable by some de ning feature These recognition theorems
usually are in terms of generators and relations esp ecially for the groups of
Lie typ e The alternating groups can also b e characterized by generators and
relations as the following theorem shows
Theorem If the group G is generated by the elements x x x
1 2 n 2
3 2 3
subject only to the relations x x for i n x x for
i i+1
1 i
2
A i n and x x for i n and i j then G
n i j
The recognition theorems for the sp oradic groups usually dep end on
how the sp oradic group was constructed For example those sp oradic groups
which were constructed by their centralizer of an involution can b e character
ized by this centralizer Theorems and are examples of recognition
theorems Those sp oradic groups which are rank are characterized by their
one p oint stabilizers See p Thus much of the discussion in Chapter two
serves to describ e some of the recognition theorems If the counterexample
group G is shown to have such characteristics as are given in the recognition
theorem of group G then the purp ose of the recognitions theorems is to state
that G is necessarily isomorphic to G
The rst two steps of the classi cation theorem are then to prove that
G has some de ning features that are in one recognition theorem We have
seen that centralizers of involutions and Sylow subgroups play an imp ortant
role in the internal structure of any simple group Many sophisticated features
of a group have b een discovered in relation to these two One of the reasons for
this is that Sylow subgroup structure dep ends on the prop erties of centraliz
ers of involutions since Sylow subgroups contain all of a groups involutions
and there is always an involution in the center of a Sylow subgroup 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 purp ose of this line of theory is to give precise descrip
tions of the way in which involutions in a Sylow subgroup are conjugate in the
group Some of the famous results are Glaub erman s Z theorem Thompson s
fusion lemma and Alp erin s fusion theorem Emb edding is another prop erty
of subgroups which develop ed into imp ortant theory What are called sig
nalizer functor metho ds grew out of the study of emb edded subgroups The
accumulation of all of the p ossible internal structures of a simple group can b e
summarized in the four part division of the main classi cation theorem pro of
The classi cation of nonconnected simple groups
The classi cation of connected simple groups of comp onent typ e
The classi cation of small simple groups of characteristic typ e
The classi cation of large simple groups of characteristic typ e
The de nitions of each of these are quite involved We have seen how
connected and nonconnected groups might arise Let us now de ne character
istic typ e
De nition X is characteristic typ e if F H is a group for every lo cal
subgroup H of X
Now F H is called the generalized tting subgroup of X and
F H L X F X
F X is the tting subgroup of X which means it is the unique largest
nilp otent subgroup of X
L X is the layer of X which means that L X is the pro duct of
all subnormal quasisimple subgroups of X with L X if no subnormal
subgroups exist
A subnormal subgroup of X is a subgroup Y such that Y Y
1
Y Y X for appropriate subgroups Y of X
2 n i
A quasisimple subgroup of X is a subgroup K such that K K K
and K Z K is simple
1 1
K K h k k j k K i where k k k k k k
This gives a glimpse of the complexity involved in pinning down the
internal structure of a simple group The four part division ab ove can actually
b e reduced to two parts that concerning noncharacteristic typ e and that
concerning characteristic typ e groups
We have taken a rather nontechnical lo ok at the enormous theorem
as Gorenstein refers to it Hop efully this will serve as at least an intro duction
to the main ob jective and some metho ds of the pro of A revision of the pro of
has b een suggested and b egun It was sp earheaded by Daniel Gorenstein
who unfortunately died in With such a large pro of to b egin with it
is generally held that completely new techniques would have to b e obtained
b efore any remarkable reduction in length could b e realized When the theorem
was nearing completion a headline in the New York Times read A Scho ol
of Theorists Works Itself Out of a Job Yet all of those involved in
the pro of had p ositive ideas of the future of group theory Gorenstein cited
applications to such elds as mathematical logic and numb er theory due to
the classi cation theorem The relationship b etween nite group theory
and nite geometries was mentioned by Aschbacher as p ossibly b ene tting
from the classi cation theorem Also even within the eld of group theory
many felt there was much to do As Gorenstein comments the obituary for
nite group theory has b een totally premature The theorem itself is a
testament to the p erseverance and co op erative nature of human kind It has
b een said in reference to the length and complexity of the theorem that either
they have b een a bit dim in nding the most e ective techniques to prove the
classi cation theorem or they have b een very clever indeed
REFERENCES
M Aschbacher The Finite Simple Groups and Their Classi
cation Yale University Press
Finite Group Theory Cambridge University Press
R W Carter Simple Groups of Lie Typ e John Wiley and Sons
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