finite simple groups - university of warwick · the classi cation of finite simple groups (cfsg)...

Finite SimpleGroups

Daniel Rogers

Why do we careabout simplegroups?

What do weknow aboutsimple groups?

What questionsare there aboutgroups in light ofthe classifcation?

The extensionproblem

Maximalsubgroups

Finite Simple Groups

Daniel Rogers

October 28 2014

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

Contents

1 Why do we care about simple groups?

2 What do we know about simple groups?

3 What questions are there about groups in light of theclassifcation?

The extension problemMaximal subgroups

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

Important note!

All groups in this talk will be finite.

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

Prime Numbers

Prime numbers are the building blocks of number theory - everyinteger can be expressed uniquely as a product of primes. As such, alot of effort goes in to understanding prime numbers.Simple groups are the equivalent notion in group theory - although,as we will see later, there are some crucial differences.

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

First attempt

Definition

A group G is too simple if the only subgroups of G are G and 1.

Theorem

The only too simple groups are cyclic groups of prime order (and thetrivial group).

This definition is, as the name suggests, too simple.

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

First attempt

Definition

Theorem

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

First attempt

Definition

Theorem

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

Key definitions

Definition

A normal subgroup N of a group G is a subgroup of G which isclosed under conjugation by elements of G ; in other words,∀ n ∈ N, g ∈ G , g−1ng ∈ N.

Definition

A group G is simple if it has precisely two normal subgroup; namelyG and 1.

Example

For p prime, the cyclic group of order p (denoted Cp) is simple.

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

Jordan-Holder theorem

Definition

A composition series for a finite group G is a chain of strict subgroups

1 = G0 < G1 < ... < Gr = G

such that each Gi is a normal subgroup of Gi+1 and the factor groupGi+1/Gi is simple.

Definition

The composition factors for a group is the set of groupsG1/G0,G2/G1, ...,Gr/Gr−1

Every (finite) group has a composition series.

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

Example (GL(2,5))

Let G = GL(2, 5), the set of invertible 2× 2 matrices over theintegers modulo 5 (this is a group of order 480). G has the followingnormal subgroups:

±SL(2, 5), the group of all matrices with determinant ±1 (order240).

SL(2, 5), the group of all matrices with determinant 1 (order120).

Z , the group of all scalar matrices (order 4).12Z , the group consisting of I , the identity, and −I (order 2).

1, the trivial group (order 1).

Then we have the following two composition series:

1 E 12Z E Z E±SL(2, 5) E G

1 E 12Z E SL(2, 5) E±SL(2, 5) E G .

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

Example (GL(2,5))

1 E 12Z E Z E±SL(2, 5) E G

1 E 12Z E SL(2, 5) E±SL(2, 5) E G .

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

Example (GL(2,5))

1 E 12Z E Z E±SL(2, 5) E G

1 E 12Z E SL(2, 5) E±SL(2, 5) E G .

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

Example (GL(2,5))

1 E 12Z E Z E±SL(2, 5) E G

1 E 12Z E SL(2, 5) E±SL(2, 5) E G .

These composition series, although different, have the same (multisetof) composition factors, namely C2,C2,C2,PSL(2, 5) ∼= Alt(5).

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

Theorem (Jordan-Holder)

Any two composition series for a group G have the same compositionfactors, up to permutation and isomorphism.

This gives us a well-defined notion of ’factors’ of a group, somewhatequivalent to the notion of primes in number theory. Thus, byunderstanding all simple groups we understand all the factors of agroup.

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

The Classification of Finite Simple Groups (CFSG)

The classification of finite simple groups is a question which tookover a century from proposal to proof.

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

CFSG history

1832 - Evariste Galois definesa normal subgroup and provesthat Alt(n) is simple forn > 4.

1870 - Camille Jordandiscovers 4 classes of simplegroups, which we now call theclassical groups, over fields ofprime order.

1873 - Emile Mathieudiscovers 5 ’sporadic’ simplegroups (ones that are notpart of infinite families).

1892 - Otto Holder first asksfor a classification of finitesimple groups.

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

CFSG history

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

CFSG history

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

CFSG history

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

CFSG history

1901 - Leonard Dicksongeneralises the classicalgroups to any finite field, anddiscovers new infinite familiesof simple groups. Moreinfinite families are found by:

Claude Chevalley (1955)Robert Steinberg (1959)Michio Suzuki (1960)Rimhak Ree (1961)

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

CFSG history

Claude Chevalley (1955)

Robert Steinberg (1959)Michio Suzuki (1960)Rimhak Ree (1961)

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

CFSG history

Claude Chevalley (1955)Robert Steinberg (1959)

Michio Suzuki (1960)Rimhak Ree (1961)

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

CFSG history

Claude Chevalley (1955)Robert Steinberg (1959)Michio Suzuki (1960)

Rimhak Ree (1961)

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

CFSG history

Claude Chevalley (1955)Robert Steinberg (1959)Michio Suzuki (1960)Rimhak Ree (1961)

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

CFSG history

1963 - Over the course of 255 pages, Walter Feit and John GriggsThompson prove a remarkable result:

Theorem (Feit-Thompson)

Every finite group of odd order is solvable (i.e. has a compositionseries whose factors are all abelian).

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

CFSG history

1966 - Zvonimir Jankodiscovers a new sporadicsimple group

1968-82 - 20 other sporadicsimple groups are discoveredby various mathematicianssuch as John Conway(pictured), Richard Lyons andMichael O’Nan.

1972 - Daniel Gorensteinproposes a 16-point plan toclassify all finite simplegroups.

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

CFSG history

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

CFSG history

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

CFSG history

1983 - Gorenstein announcesthat the classification iscomplete!

1997 - Michael Aschbacherannounces that it isn’t.

2004 - Aschbacher andStephen Smith fill in the gap(across 1221 pages)

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

CFSG history

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

CFSG history

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

CFSG today

This first proof took about 10,000 pages, with over 100 authors.

Work began almost immediately on a ”second generation”proof. This is ongoing and expected to fill about 5,000 pages

Some work has also started on a ”third generation” proof.

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

The Classification of Finite Simple Groups (CFSG)

Theorem

Every finite simple group is isomorphic to one of the followinggroups:

Cp for p prime

Alt(n) for n ≥ 5.

A simple group of Lie type

One of the 26 sporadic simple groups

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

Groups of Lie type

”Most” finite simple groups fall into this category. These are fullyclassified using Dynkin diagrams.

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

The classical groups - linear

PSL(n, q), for n ∈ Z>1, q = pe , p prime.

Start with GL(n, q), the set of all n × n matrices with entries inthe finite field F with q elements. (For instance if q = p thenF = Zp)

Take the subgroup SL(n, q) of all matrices with determinant 1.

Obtain PSL(n, q) by quotienting out by all scalar matrices withdeterminant 1.

This group is simple in all except a couple of small cases.

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

The classical groups - unitary

PSU(n, q), for n ∈ Z>1, q = pe , p prime.

Start with GL(n, q2), the set of all n× n matrices with entries inthe finite field F with q2 elements.

Construct the map¯: F→ F, f = f q. (Think of this map asbeing like complex conjugation). Extend this map to GL(n, q2),by sending A = (ai,j) to A = (ai,j).

Look at the subgroup GU(n, q) of GL(n, q2) consisting of

matrices A such that AAT

= In. (Such matrices are calledunitary matrices.)

Take the subgroup SU(n, q) of all such matrices withdeterminant 1.

Obtain PSU(n, q) by quotienting out by all scalar matrices withdeterminant 1.

This group is simple in all except a few small cases.

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

The classical groups - symplectic

PSp(2n, q), for n ∈ Z>1, q = pe , p prime.

Start with GL(2n, q), the set of all 2n× 2n matrices with entriesin the finite field F with q elements.

Let F = antidiag(−1,−1, ...,−1︸︷︷︸n times

, 1, 1, ..., 1︸︷︷︸n times

), and consider the

subgroup Sp(2n, q) of GL(2n, q) of matrices A such thatAFAT = F .

It turns out that such matrices always have determinant 1.

Obtain PSp(2n, q) by quotienting out by all scalar matrices withdeterminant 1.

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

, 1, 1, ..., 1︸︷︷︸n times

), and consider the

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

, 1, 1, ..., 1︸︷︷︸n times

), and consider the

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

, 1, 1, ..., 1︸︷︷︸n times

), and consider the

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

The classical groups - orthogonal

POε(n, q), for n ∈ Z>1, q = pe , p 6= 2 prime.

Start with GL(n, q), the set of all n × n matrices with entries inthe finite field F with q elements.

Let F be a symmetric matrix (so FT = F ), and consider thesubgroup SO(n, q) of SL(n, q) of matrices A of determinant 1such that AFAT = F .

It turns out that this group SO(n, q) has a (unique) subgroup ofindex 2, called Ω(n, q). (This is often defined as the kernel of acertain homomorphism called the spinor norm).

Obtain PO(n, q) by quotienting out by all scalar matrices withdeterminant 1.

If n is odd, then regardless of our choice of form all such groupswill be isomorphic, and are usually denoted PO.

If n is even, then we get two different groups, denoted PO+ andPO−.

These groups are simple in all except a few small cases.

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

Sporadic simple groups

These are 26 ’other’ groups, that don’t fall into any other families.(The fact that these groups exist is one reason why the classificationis so difficult).

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

The Monster group

Normally denoted M, this was constructed by Robert Greiss in 1982.

Has order808017424794512875886459904961710757005754368000000000.

It can be defined as a subgroup of Sym(n) but the smallest suchn is approximately 1020.

It can also be defined as a matrix group in 196,883 dimensions.(Or 196,882 in characteristic 2).

It contains at least 20 of the 26 sporadic groups.

This is one of the hardest simple groups to work with becausethere is no ”nice” way to look at it.

For instance, Alt(100) is of much larger order, but because wecan define this as permutations on 100 points it is much easierto deal with.

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

The Monster group

Has order808017424794512875886459904961710757005754368000000000.

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

The Monster group

Has order808017424794512875886459904961710757005754368000000000.

It can be defined as a subgroup of Sym(n)

but the smallest suchn is approximately 1020.

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

The Monster group

Has order808017424794512875886459904961710757005754368000000000.

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

The Monster group

Has order808017424794512875886459904961710757005754368000000000.

It can also be defined as a matrix group

in 196,883 dimensions.(Or 196,882 in characteristic 2).

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

The Monster group

Has order808017424794512875886459904961710757005754368000000000.

It can also be defined as a matrix group in 196,883 dimensions.

(Or 196,882 in characteristic 2).

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

The Monster group

Has order808017424794512875886459904961710757005754368000000000.

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

The Monster group

Has order808017424794512875886459904961710757005754368000000000.

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

The Monster group

Has order808017424794512875886459904961710757005754368000000000.

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

The problem with these factors

As discussed, the simple groups act for group theory like primenumbers do for number theory. However there is a problem:

Number Theory Group Theory

Classify all ‘primes’ HARD HARD BUT DONE

Construct elementsfrom their

‘prime factors’EASY HARD AND NOT DONE

Numbers have unique factorisation, so it is very easy to construct theunique number with a given factorisation. For groups this isn’tanything like as straightforward.

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

Classify all ‘primes’ HARD

HARD BUT DONE

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

‘prime factors’

EASY HARD AND NOT DONE

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

‘prime factors’EASY

HARD AND NOT DONE

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

Example - Alt(6)

Example

Let G = 〈(1, 2)(3, 4)(5, 6)(7, 8), (2, 4, 8, 9)(3, 7, 6, 0)〉. In factG ∼= Alt(6), and so |G | = 360. There are various ways to add anelement to G to give us a group of order 720.

Adding s1 := (1, 3, 7, 4, 2, 8)(5, 0, 9) gives Sym(6).

Adding s2 := (1, 8, 4, 6, 2, 7, 3, 5) gives PGL(2, 9).

Adding s1s2 gives M10, the stabiliser of a single point in thesporadic simple group M11.

These all have the same composition factors, namely Alt(6),C2,but none of these three groups are isomorphic.

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

Example - Alt(6)

Example

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

Example - Alt(6)

Example

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

Example - Alt(6)

Example

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

Example - Alt(6)

Example

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

The extension problem

Given two abstract groups H and N, we can ask for a classification ofall groups G such that G has a normal subgroup N and G/N ∼= H.Here G is called an extension of H by N, and finding thisclassification is known as the extension problem.

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

Types of extension

Some groups with composition factors Alt(6),C2 are:

The direct product Alt(6)× C2.

A semidirect product (sometimes called a split extension)Alt(6)o C2. All three of the groups described earlier fall intothis category (and these are the only such ones in this case).

A central extension - this is a group G such that H is containedin the center of G , and G/H ∼= K . There is a perfect centralextension of Alt(6) by C2, usually denoted 2.Alt(6). Here this isSL(2, 9).

This is all possible groups with this structure.

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

Types of extension

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

Types of extension

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

Types of extension

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

The extension problem

The extension problem is hard and currently unsolved! Understandingthis problem would allow us to produce the Classification of FiniteGroups.

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

Maximal Subgroups

Definition

A subgroup H < G is maximal if H 6= G and there is no subgroup Msuch that H M G .

Example

Let G = Alt(5) = 〈(1, 2)(3, 4), (1, 2, 3, 4, 5)〉 of order 60. Then itsmaximal subgroups (up to conjugacy) are:

Alt(4) ∼= 〈(1, 2)(3, 4), (1, 2, 3)〉, order 12.

D5∼= 〈(1, 2, 3, 4, 5), (1, 5)(2, 4)〉, order 10.

Sym(3) ∼= 〈(1, 2, 3), (2, 3)(4, 5)〉, order 6.

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

Maximal Subgroups

Definition

A subgroup H < G is maximal if H 6= G and there is no subgroup Msuch that H M G .

Example

Let G = Alt(5) = 〈(1, 2)(3, 4), (1, 2, 3, 4, 5)〉 of order 60. Then itsmaximal subgroups (up to conjugacy) are:

Alt(4) ∼= 〈(1, 2)(3, 4), (1, 2, 3)〉, order 12.

D5∼= 〈(1, 2, 3, 4, 5), (1, 5)(2, 4)〉, order 10.

Sym(3) ∼= 〈(1, 2, 3), (2, 3)(4, 5)〉, order 6.

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

Why are maximal subgroups useful?

Understanding maximal subgroups allows us to understand allsubgroups.

Example

Alt(5) has no subgroup of order 15.

There is a 1:1 correspondence between maximal subgroups of asimple group, and primitive permutation groups isomorphic tothem. So understanding maximal subgroups allows us to write agiven group in various different ways as permutation groups.

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

Example

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

Example

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

Example

The maximal subgroups of Alt(5) (a group of order 60) are Alt(4)(order 12), D5 (order 10) and Sym(3) (order 6). These correspond tothree different permutation representations of Alt(5):

Alt(4) gives us 〈(1, 2, 3, 4, 5), (1, 2)(3, 4)〉.D5 gives us 〈(1, 3, 4)(2, 5, 6), (2, 6, 3, 5, 4)〉.Sym(3) gives us〈(1, 2, 3, 4, 5)(6, 7, 8, 9, 0), (1, 2)(5, 0)(4, 7)(3, 9)〉.

These are all the ”genuinely different” permutation representations ofAlt(5) (without being precise as to what I mean by ”genuinelydifferent”)

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

The O’Nan-Scott theorem

Discovered independently by O’Nan and Scott, this classifies themaximal subgroups of the symmetric group.

Theorem

The maximal subgroups of Sym(n) is contained in (at least one of):

Sk × Sn−k for some k, 0 < k < n.

A wreath product Sk o Sl where kl = n.

A selection of specific primitive groups

Proof.

Relatively unpleasant (but not too long). See for example The FiniteSimple Groups by Wilson.

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

Aschbacher’s Theorem

Aschbacher gives a classification of maximal subgroups of classicalgroups (recall earlier) into 9 classes, denoted C1, ...,C9, althoughthese classes can’t be described in a slide very easily.

Classes C1, ...,C8 have geometric structure, and these classeshave been classified completely by Kleidman and Liebeck.

C9 is essentially ”everything else” and requires more case by caseanalysis. In particular these generally have to be studied inindividual dimensions and it is very unlikely that a fullclassification of all the groups in this class will be possible.

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

Class C9

A lot of the groups that occur in this class relate to the irreduciblerepresentations of various simple groups. If a group has an irreduciblecharacter of degree n then this group will exist as a subgroup ofSL(n, q) where q = pe (and possibly inside another classical group)and, with some technicalities, these are often maximal.

Finite SimpleGroups

Daniel Rogers

Maximalsubgroups

Classifying C9 subgroups

Dimensions up to 12 done by Bray, Holt and Roney-Dougal(2013).

Dimensions 13, 14 and 15 work in progress by Schroeder.

Dimensions 16 and 17 work in progress by R .

finite simple groups - university of warwick · the classi cation of finite simple groups (cfsg)...

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