Finite Simple Groups

Daniel Rogers

Why do we care about simple groups?

What do we know about simple groups? Finite Simple Groups

What questions are there about groups in light of the classifcation? The extension Daniel Rogers problem Maximal subgroups

October 28 2014 Contents

Finite Simple Groups

Daniel Rogers

Why do we care about simple groups? 1 Why do we care about simple groups? What do we know about simple groups? What questions 2 are there about What do we know about simple groups? groups in light of the classifcation? The extension problem Maximal 3 What questions are there about groups in light of the subgroups classifcation? The extension problem Maximal subgroups Important note!

Finite Simple Groups

Daniel Rogers

Why do we care about simple groups?

What do we know about simple groups?

What questions are there about All groups in this talk will be ﬁnite. groups in light of the classifcation? The extension problem Maximal subgroups Prime Numbers

Finite Simple Groups

Daniel Rogers

Why do we care about simple groups?

What do we know about simple groups? Prime numbers are the building blocks of number theory - every What questions integer can be expressed uniquely as a product of primes. As such, a are there about groups in light of lot of eﬀort goes in to understanding prime numbers. the classifcation? The extension Simple groups are the equivalent notion in group theory - although, problem Maximal as we will see later, there are some crucial diﬀerences. subgroups Theorem The only too simple groups are cyclic groups of prime order (and the trivial group).

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

First attempt

Finite Simple Groups

Daniel Rogers

Why do we care about simple groups?

What do we Deﬁnition know about simple groups? A group G is too simple if the only subgroups of G are G and 1. What questions are there about groups in light of the classifcation? The extension problem Maximal subgroups This deﬁnition is, as the name suggests, too simple.

First attempt

Finite Simple Groups

Daniel Rogers

Why do we care about simple groups?

What do we Deﬁnition know about simple groups? A group G is too simple if the only subgroups of G are G and 1. What questions are there about groups in light of the classifcation? Theorem The extension problem The only too simple groups are cyclic groups of prime order (and the Maximal subgroups trivial group). First attempt

Finite Simple Groups

Daniel Rogers

Why do we care about simple groups?

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

Finite Simple Groups

Daniel Rogers

Why do we care Deﬁnition about simple groups? A normal subgroup N of a group G is a subgroup of G which is What do we know about closed under conjugation by elements of G; in other words, simple groups? ∀ n ∈ N, g ∈ G, g −1ng ∈ N. What questions are there about groups in light of the classifcation? Deﬁnition The extension problem Maximal A group G is simple if it has precisely two normal subgroup; namely subgroups G and 1.

Example

For p prime, the cyclic group of order p (denoted Cp) is simple. Jordan-H¨oldertheorem

Finite Simple Groups

Daniel Rogers

Why do we care Definition about simple groups? A composition series for a finite group G is a chain of strict subgroups What do we know about simple groups? 1 = G0 < G1 < ... < Gr = G What questions are there about groups in light of such that each Gi is a normal subgroup of Gi+1 and the factor group the classifcation? The extension Gi+1/Gi is simple. problem Maximal subgroups Definition The composition factors for a group is the set of groups {G1/G0, G2/G1, ..., Gr /Gr−1} Every (finite) group has a composition series. ±SL(2, 5), the group of all matrices with determinant ±1 (order 240). SL(2, 5), the group of all matrices with determinant 1 (order 120). Z, the group of all scalar matrices (order 4). 1 2 Z, 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 1 E 2 Z E Z E ±SL(2, 5) E G 1 1 E 2 Z E SL(2, 5) E ±SL(2, 5) E G.

Jordan-H¨oldertheorem

Finite Simple Groups Example (GL(2,5)) Daniel Rogers Let G = GL(2, 5), the set of invertible 2 × 2 matrices over the Why do we care about simple integers modulo 5 (this is a group of order 480). G has the following groups?

What do we normal subgroups: know about simple groups?

What questions are there about groups in light of the classifcation? The extension problem Maximal subgroups Then we have the following two composition series: 1 1 E 2 Z E Z E ±SL(2, 5) E G 1 1 E 2 Z E SL(2, 5) E ±SL(2, 5) E G.

Jordan-H¨oldertheorem

What do we normal subgroups: know about simple groups? ±SL(2, 5), the group of all matrices with determinant ±1 (order What questions 240). are there about groups in light of the classifcation? SL(2, 5), the group of all matrices with determinant 1 (order The extension problem 120). Maximal subgroups Z, the group of all scalar matrices (order 4). 1 2 Z, the group consisting of I , the identity, and −I (order 2). 1, the trivial group (order 1). Jordan-H¨oldertheorem

Finite Simple Groups

Daniel Rogers

Why do we care about simple groups?

What do we know about Example (GL(2,5)) simple groups?

What questions 1 are there about 1 E 2 Z E Z E ±SL(2, 5) E G groups in light of 1 the classifcation? 1 E 2 Z E SL(2, 5) E ±SL(2, 5) E G. The extension problem Maximal These composition series, although diﬀerent, have the same (multiset subgroups ∼ of) composition factors, namely {C2, C2, C2, PSL(2, 5) = Alt(5)}. Jordan-H¨oldertheorem

Finite Simple Groups

Daniel Rogers

Why do we care about simple groups?

What do we Theorem (Jordan-Hölder) know about simple groups? Any two composition series for a group G have the same composition What questions are there about factors, up to permutation and isomorphism. groups in light of the classifcation? The extension This gives us a well-defined notion of ’factors’ of a group, somewhat problem Maximal equivalent to the notion of primes in number theory. Thus, by subgroups understanding all simple groups we understand all the factors of a group. The Classification of Finite Simple Groups (CFSG)

Finite Simple Groups

Daniel Rogers

Why do we care about simple groups?

What do we know about simple groups?

What questions are there about The classification of finite simple groups is a question which took groups in light of the classifcation? over a century from proposal to proof. The extension problem Maximal subgroups 1870 - Camille Jordan discovers 4 classes of simple groups, which we now call the classical groups, over fields of prime order. 1873 - Emile´ Mathieu discovers 5 ’sporadic’ simple groups (ones that are not part of infinite families). 1892 - Otto Hölderfirst asks for a classification of finite simple groups.

CFSG history

Finite Simple Groups Daniel Rogers 1832 - Evariste´ Galois deﬁnes

Why do we care a normal subgroup and proves about simple groups? that Alt(n) is simple for

What do we n > 4. know about simple groups?

What questions are there about groups in light of the classifcation? The extension problem Maximal subgroups 1873 - Emile´ Mathieu discovers 5 ’sporadic’ simple groups (ones that are not part of infinite families). 1892 - Otto Hölderfirst asks for a classification of finite simple groups.

CFSG history

Finite Simple Groups Daniel Rogers 1832 - Evariste´ Galois deﬁnes

Why do we care a normal subgroup and proves about simple groups? that Alt(n) is simple for

CFSG history

Finite Simple Groups Daniel Rogers 1832 - Evariste´ Galois deﬁnes

Why do we care a normal subgroup and proves about simple groups? that Alt(n) is simple for

What do we n > 4. know about simple groups? 1870 - Camille Jordan What questions are there about discovers 4 classes of simple groups in light of the classifcation? groups, which we now call the The extension problem classical groups, over ﬁelds of Maximal subgroups prime order. 1873 - Emile´ Mathieu discovers 5 ’sporadic’ simple groups (ones that are not part of inﬁnite families). CFSG history

Finite Simple Groups Daniel Rogers 1832 - Evariste´ Galois deﬁnes

Why do we care a normal subgroup and proves about simple groups? that Alt(n) is simple for

CFSG history

Finite Simple Groups Daniel Rogers 1901 - Leonard Dickson generalises the classical Why do we care about simple groups to any finite field, and groups? discovers new infinite families What do we know about of simple groups. More simple groups? infinite families are found by: What questions are there about groups in light of the classifcation? The extension problem Maximal subgroups Robert Steinberg (1959) Michio Suzuki (1960) Rimhak Ree (1961)

CFSG history

Finite Simple Groups Daniel Rogers 1901 - Leonard Dickson generalises the classical Why do we care about simple groups to any finite field, and groups? discovers new infinite families What do we know about of simple groups. More simple groups? infinite families are found by: What questions are there about Claude Chevalley (1955) groups in light of the classifcation? The extension problem Maximal subgroups Michio Suzuki (1960) Rimhak Ree (1961)

CFSG history

Finite Simple Groups Daniel Rogers 1901 - Leonard Dickson generalises the classical Why do we care about simple groups to any finite field, and groups? discovers new infinite families What do we know about of simple groups. More simple groups? infinite families are found by: What questions are there about Claude Chevalley (1955) groups in light of the classifcation? Robert Steinberg (1959) The extension problem Maximal subgroups Rimhak Ree (1961)

CFSG history

Finite Simple Groups Daniel Rogers 1901 - Leonard Dickson Why do we care generalises the classical about simple groups? groups to any finite field, and What do we discovers new infinite families know about simple groups? of simple groups. More

What questions inﬁnite families are found by: are there about groups in light of Claude Chevalley (1955) the classifcation? The extension Robert Steinberg (1959) problem Maximal Michio Suzuki (1960) subgroups CFSG history

Finite Simple Groups

Daniel Rogers

Why do we care about simple groups?

What do we know about simple groups?

What questions are there about groups in light of the classifcation? The extension problem Maximal subgroups 1963 - Over the course of 255 pages, Walter Feit and John Griggs Thompson prove a remarkable result: Theorem (Feit-Thompson) Every ﬁnite group of odd order is solvable (i.e. has a composition series whose factors are all abelian). 1968-82 - 20 other sporadic simple groups are discovered by various mathematicians such as John Conway (pictured), Richard Lyons and Michael O’Nan. 1972 - Daniel Gorenstein proposes a 16-point plan to classify all ﬁnite simple groups.

CFSG history

Finite Simple Groups Daniel Rogers 1966 - Zvonimir Janko

Why do we care discovers a new sporadic about simple groups? simple group

What do we know about simple groups?

What questions are there about groups in light of the classifcation? The extension problem Maximal subgroups 1972 - Daniel Gorenstein proposes a 16-point plan to classify all ﬁnite simple groups.

CFSG history

Finite Simple Groups

Daniel Rogers 1966 - Zvonimir Janko Why do we care about simple discovers a new sporadic groups? simple group What do we know about 1968-82 - 20 other sporadic simple groups? simple groups are discovered What questions are there about by various mathematicians groups in light of the classifcation? such as John Conway The extension problem (pictured), Richard Lyons and Maximal subgroups Michael O’Nan. 1972 - Daniel Gorenstein proposes a 16-point plan to classify all ﬁnite simple groups. 1997 - Michael Aschbacher announces that it isn’t. 2004 - Aschbacher and Stephen Smith ﬁll in the gap (across 1221 pages)

CFSG history

Finite Simple Groups Daniel Rogers 1983 - Gorenstein announces

Why do we care that the classiﬁcation is about simple groups? complete!

What do we know about simple groups?

What questions are there about groups in light of the classifcation? The extension problem Maximal subgroups 2004 - Aschbacher and Stephen Smith ﬁll in the gap (across 1221 pages)

CFSG history

Finite Simple Groups

Daniel Rogers 1983 - Gorenstein announces Why do we care about simple that the classiﬁcation is groups? complete! What do we know about 1997 - Michael Aschbacher simple groups? announces that it isn’t. What questions are there about groups in light of the classifcation? The extension problem Maximal subgroups CFSG history

Finite Simple Groups

Daniel Rogers 1983 - Gorenstein announces Why do we care about simple that the classiﬁcation is groups? complete! What do we know about 1997 - Michael Aschbacher simple groups? announces that it isn’t. What questions are there about groups in light of 2004 - Aschbacher and the classifcation? The extension Stephen Smith ﬁll in the gap problem Maximal (across 1221 pages) subgroups CFSG today

Finite Simple Groups

Daniel Rogers

Why do we care about simple groups?

What do we know about simple groups? This first proof took about 10,000 pages, with over 100 authors. What questions are there about Work began almost immediately on a ”second generation” groups in light of the classifcation? proof. This is ongoing and expected to fill about 5,000 pages The extension problem Some work has also started on a ”third generation” proof. Maximal subgroups The Classification of Finite Simple Groups (CFSG)

Finite Simple Groups

Daniel Rogers

Why do we care about simple groups? Theorem What do we know about simple groups? Every ﬁnite simple group is isomorphic to one of the following

What questions groups: are there about groups in light of C for p prime the classifcation? p The extension problem Alt(n) for n ≥ 5. Maximal subgroups A simple group of Lie type One of the 26 sporadic simple groups Groups of Lie type

Finite Simple Groups

Daniel Rogers

Why do we care about simple ”Most” ﬁnite simple groups fall into this category. These are fully groups?

What do we classiﬁed using Dynkin diagrams. know about simple groups?

What questions are there about groups in light of the classifcation? The extension problem Maximal subgroups Take the subgroup SL(n, q) of all matrices with determinant 1. Obtain PSL(n, q) by quotienting out by all scalar matrices with determinant 1. This group is simple in all except a couple of small cases.

The classical groups - linear

Finite Simple Groups

Daniel Rogers

The classical groups - linear

Finite Simple Groups

Daniel Rogers

Finite Simple Groups

Daniel Rogers

Why do we care about simple e groups? PSL(n, q), for n ∈ Z>1, q = p , p prime. What do we know about Start with GL(n, q), the set of all n × n matrices with entries in simple groups? the ﬁnite ﬁeld with q elements. (For instance if q = p then What questions F are there about = p) groups in light of F Z the classifcation? Take the subgroup SL(n, q) of all matrices with determinant 1. The extension problem Maximal Obtain PSL(n, q) by quotienting out by all scalar matrices with subgroups determinant 1. This group is simple in all except a couple of small cases. Construct the map¯: F → F, f = f q. (Think of this map as being 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 T matrices A such that AA = In. (Such matrices are called unitary matrices.) Take the subgroup SU(n, q) of all such matrices with determinant 1. Obtain PSU(n, q) by quotienting out by all scalar matrices with determinant 1. This group is simple in all except a few small cases.

The classical groups - unitary

What do we know about simple groups?

What questions are there about groups in light of the classifcation? The extension problem Maximal subgroups Look at the subgroup GU(n, q) of GL(n, q2) consisting of T matrices A such that AA = In. (Such matrices are called unitary matrices.) Take the subgroup SU(n, q) of all such matrices with determinant 1. Obtain PSU(n, q) by quotienting out by all scalar matrices with determinant 1. This group is simple in all except a few small cases.

The classical groups - unitary

Finite Simple Groups e Daniel Rogers PSU(n, q), for n ∈ Z>1, q = p , p prime. Start with GL(n, q2), the set of all n × n matrices with entries in Why do we care about simple the ﬁnite ﬁeld with q2 elements. groups? F q What do we Construct the map¯: F → F, f = f . (Think of this map as know about 2 simple groups? being like complex conjugation). Extend this map to GL(n, q ), What questions by sending A = (ai,j ) to A = (ai,j ). are there about groups in light of 2 the classifcation? Look at the subgroup GU(n, q) of GL(n, q ) consisting of The extension T problem matrices A such that AA = In. (Such matrices are called Maximal subgroups unitary matrices.) Take the subgroup SU(n, q) of all such matrices with determinant 1. Obtain PSU(n, q) by quotienting out by all scalar matrices with determinant 1. This group is simple in all except a few small cases. Let F = antidiag(−1, −1, ..., −1, 1, 1, ..., 1), and consider the | {z } | {z } n times n times subgroup Sp(2n, q) of GL(2n, q) of matrices A such that AFAT = F . It turns out that such matrices always have determinant 1. Obtain PSp(2n, q) by quotienting out by all scalar matrices with determinant 1. This group is simple in all except a few small cases.

The classical groups - symplectic

Finite Simple Groups

What questions are there about groups in light of the classifcation? The extension problem Maximal subgroups It turns out that such matrices always have determinant 1. Obtain PSp(2n, q) by quotienting out by all scalar matrices with determinant 1. This group is simple in all except a few small cases.

The classical groups - symplectic

Finite Simple Groups

The classical groups - symplectic

Finite Simple Groups

Daniel Rogers e Why do we care PSp(2n, q), for n ∈ Z>1, q = p , p prime. about simple groups? Start with GL(2n, q), the set of all 2n × 2n matrices with entries What do we in the finite field with q elements. know about F simple groups? Let F = antidiag(−1, −1, ..., −1, 1, 1, ..., 1), and consider the What questions are there about | {z } | {z } groups in light of n times n times the classifcation? subgroup Sp(2n, q) of GL(2n, q) of matrices A such that The extension T problem AFA = F . Maximal subgroups It turns out that such matrices always have determinant 1. Obtain PSp(2n, q) by quotienting out by all scalar matrices with determinant 1. This group is simple in all except a few small cases. Let F be a symmetric matrix (so F T = F ), and consider the subgroup SO(n, q) of SL(n, q) of matrices A of determinant 1 such that AFAT = F . It turns out that this group SO(n, q) has a (unique) subgroup of index 2, called Ω(n, q). (This is often defined as the kernel of a certain homomorphism called the spinor norm). Obtain PO(n, q) by quotienting out by all scalar matrices with determinant 1. If n is odd, then regardless of our choice of form all such groups will be isomorphic, and are usually denoted PO◦. If n is even, then we get two different groups, denoted PO+ and PO−. These groups are simple in all except a few small cases.

The classical groups - orthogonal

Finite Simple e Groups PO (n, q), for n ∈ Z>1, q = p , p 6= 2 prime. Daniel Rogers Start with GL(n, q), the set of all n × n matrices with entries in

Why do we care the ﬁnite ﬁeld F with q elements. about simple groups?

What do we know about simple groups?

What questions are there about groups in light of the classifcation? The extension problem Maximal subgroups It turns out that this group SO(n, q) has a (unique) subgroup of index 2, called Ω(n, q). (This is often deﬁned as the kernel of a certain homomorphism called the spinor norm). Obtain PO(n, q) by quotienting out by all scalar matrices with determinant 1. If n is odd, then regardless of our choice of form all such groups will be isomorphic, and are usually denoted PO◦. If n is even, then we get two diﬀerent groups, denoted PO+ and PO−. These groups are simple in all except a few small cases.

The classical groups - orthogonal

Finite Simple e Groups PO (n, q), for n ∈ Z>1, q = p , p 6= 2 prime. Daniel Rogers Start with GL(n, q), the set of all n × n matrices with entries in

Why do we care the finite field F with q elements. about simple T groups? Let F be a symmetric matrix (so F = F ), and consider the What do we know about subgroup SO(n, q) of SL(n, q) of matrices A of determinant 1 simple groups? such that AFAT = F . What questions are there about groups in light of the classifcation? The extension problem Maximal subgroups Obtain PO(n, q) by quotienting out by all scalar matrices with determinant 1. If n is odd, then regardless of our choice of form all such groups will be isomorphic, and are usually denoted PO◦. If n is even, then we get two different groups, denoted PO+ and PO−. These groups are simple in all except a few small cases.

The classical groups - orthogonal

Finite Simple e Groups PO (n, q), for n ∈ Z>1, q = p , p 6= 2 prime. Daniel Rogers Start with GL(n, q), the set of all n × n matrices with entries in

Why do we care the finite field F with q elements. about simple T groups? Let F be a symmetric matrix (so F = F ), and consider the What do we know about subgroup SO(n, q) of SL(n, q) of matrices A of determinant 1 simple groups? such that AFAT = F . What questions are there about It turns out that this group SO(n, q) has a (unique) subgroup of groups in light of the classifcation? index 2, called Ω(n, q). (This is often defined as the kernel of a The extension problem certain homomorphism called the spinor norm). Maximal subgroups If n is odd, then regardless of our choice of form all such groups will be isomorphic, and are usually denoted PO◦. If n is even, then we get two different groups, denoted PO+ and PO−. These groups are simple in all except a few small cases.

The classical groups - orthogonal

Finite Simple e Groups PO (n, q), for n ∈ Z>1, q = p , p 6= 2 prime. Daniel Rogers Start with GL(n, q), the set of all n × n matrices with entries in

Why do we care the finite field F with q elements. about simple T groups? Let F be a symmetric matrix (so F = F ), and consider the What do we know about subgroup SO(n, q) of SL(n, q) of matrices A of determinant 1 simple groups? such that AFAT = F . What questions are there about It turns out that this group SO(n, q) has a (unique) subgroup of groups in light of the classifcation? index 2, called Ω(n, q). (This is often defined as the kernel of a The extension problem certain homomorphism called the spinor norm). Maximal subgroups Obtain PO(n, q) by quotienting out by all scalar matrices with determinant 1. If n is even, then we get two different groups, denoted PO+ and PO−. These groups are simple in all except a few small cases.

The classical groups - orthogonal

Finite Simple e Groups PO (n, q), for n ∈ Z>1, q = p , p 6= 2 prime. Daniel Rogers Start with GL(n, q), the set of all n × n matrices with entries in

Why do we care the finite field F with q elements. about simple T groups? Let F be a symmetric matrix (so F = F ), and consider the What do we know about subgroup SO(n, q) of SL(n, q) of matrices A of determinant 1 simple groups? such that AFAT = F . What questions are there about It turns out that this group SO(n, q) has a (unique) subgroup of groups in light of the classifcation? index 2, called Ω(n, q). (This is often defined as the kernel of a The extension problem certain homomorphism called the spinor norm). Maximal subgroups Obtain PO(n, q) by quotienting out by all scalar matrices with determinant 1. If n is odd, then regardless of our choice of form all such groups will be isomorphic, and are usually denoted PO◦. The classical groups - orthogonal

Finite Simple e Groups PO (n, q), for n ∈ Z>1, q = p , p 6= 2 prime. Daniel Rogers Start with GL(n, q), the set of all n × n matrices with entries in

Why do we care the finite field F with q elements. about simple T groups? Let F be a symmetric matrix (so F = F ), and consider the What do we know about subgroup SO(n, q) of SL(n, q) of matrices A of determinant 1 simple groups? such that AFAT = F . What questions are there about It turns out that this group SO(n, q) has a (unique) subgroup of groups in light of the classifcation? index 2, called Ω(n, q). (This is often defined as the kernel of a The extension problem certain homomorphism called the spinor norm). Maximal subgroups Obtain PO(n, q) by quotienting out by all scalar matrices with determinant 1. If n is odd, then regardless of our choice of form all such groups will be isomorphic, and are usually denoted PO◦. If n is even, then we get two different groups, denoted PO+ and PO−. These groups are simple in all except a few small cases. Sporadic simple groups

Finite Simple Groups

Daniel Rogers

Why do we care about simple groups?

What do we know about simple groups?

What questions These are 26 ’other’ groups, that don’t fall into any other families. are there about groups in light of (The fact that these groups exist is one reason why the classification the classifcation? is so difficult). The extension problem Maximal subgroups Has order 808017424794512875886459904961710757005754368000000000. It can be defined as a subgroup of Sym(n) but the smallest such n 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 because there is no ”nice” way to look at it. For instance, Alt(100) is of much larger order, but because we can define this as permutations on 100 points it is much easier to deal with.

The Monster group

Finite Simple Groups Daniel Rogers Normally denoted M, this was constructed by Robert Greiss in 1982.

Why do we care about simple groups?

What do we know about simple groups?

What questions are there about groups in light of the classifcation? The extension problem Maximal subgroups It can be defined as a subgroup of Sym(n) but the smallest such n 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 because there is no ”nice” way to look at it. For instance, Alt(100) is of much larger order, but because we can define this as permutations on 100 points it is much easier to deal with.

The Monster group

Finite Simple Groups Daniel Rogers Normally denoted M, this was constructed by Robert Greiss in 1982.

Why do we care about simple Has order groups? 808017424794512875886459904961710757005754368000000000. What do we know about simple groups?

What questions are there about groups in light of the classifcation? The extension problem Maximal subgroups but the smallest such n 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 because there is no ”nice” way to look at it. For instance, Alt(100) is of much larger order, but because we can deﬁne this as permutations on 100 points it is much easier to deal with.

The Monster group

Finite Simple Groups Daniel Rogers Normally denoted M, this was constructed by Robert Greiss in 1982.

Why do we care about simple Has order groups? 808017424794512875886459904961710757005754368000000000. What do we know about It can be deﬁned as a subgroup of Sym(n) simple groups?

What questions are there about groups in light of the classifcation? The extension problem Maximal subgroups 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 because there is no ”nice” way to look at it. For instance, Alt(100) is of much larger order, but because we can deﬁne this as permutations on 100 points it is much easier to deal with.

The Monster group

Finite Simple Groups Daniel Rogers Normally denoted M, this was constructed by Robert Greiss in 1982.

Why do we care about simple Has order groups? 808017424794512875886459904961710757005754368000000000. What do we know about It can be deﬁned as a subgroup of Sym(n) but the smallest such simple groups? n is approximately 1020. What questions are there about groups in light of the classifcation? The extension problem Maximal subgroups 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 because there is no ”nice” way to look at it. For instance, Alt(100) is of much larger order, but because we can deﬁne this as permutations on 100 points it is much easier to deal with.

The Monster group

Finite Simple Groups Daniel Rogers Normally denoted M, this was constructed by Robert Greiss in 1982.

Why do we care about simple Has order groups? 808017424794512875886459904961710757005754368000000000. What do we know about It can be defined as a subgroup of Sym(n) but the smallest such simple groups? n is approximately 1020. What questions are there about groups in light of It can also be defined as a matrix group the classifcation? The extension problem Maximal subgroups (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 because there is no ”nice” way to look at it. For instance, Alt(100) is of much larger order, but because we can define this as permutations on 100 points it is much easier to deal with.

The Monster group

Finite Simple Groups Daniel Rogers Normally denoted M, this was constructed by Robert Greiss in 1982.

Why do we care about simple Has order groups? 808017424794512875886459904961710757005754368000000000. What do we know about It can be defined as a subgroup of Sym(n) but the smallest such simple groups? n is approximately 1020. What questions are there about groups in light of It can also be defined as a matrix group in 196,883 dimensions. the classifcation? The extension problem Maximal subgroups It contains at least 20 of the 26 sporadic groups. This is one of the hardest simple groups to work with because there is no ”nice” way to look at it. For instance, Alt(100) is of much larger order, but because we can define this as permutations on 100 points it is much easier to deal with.

The Monster group

Finite Simple Groups Daniel Rogers Normally denoted M, this was constructed by Robert Greiss in 1982.

Why do we care about simple Has order groups? 808017424794512875886459904961710757005754368000000000. What do we know about It can be defined as a subgroup of Sym(n) but the smallest such simple groups? n is approximately 1020. What questions are there about groups in light of It can also be defined as a matrix group in 196,883 dimensions. the classifcation? The extension (Or 196,882 in characteristic 2). problem Maximal subgroups This is one of the hardest simple groups to work with because there is no ”nice” way to look at it. For instance, Alt(100) is of much larger order, but because we can define this as permutations on 100 points it is much easier to deal with.

The Monster group

Finite Simple Groups Daniel Rogers Normally denoted M, this was constructed by Robert Greiss in 1982.

Why do we care about simple Has order groups? 808017424794512875886459904961710757005754368000000000. What do we know about It can be defined as a subgroup of Sym(n) but the smallest such simple groups? n is approximately 1020. What questions are there about groups in light of It can also be defined as a matrix group in 196,883 dimensions. the classifcation? The extension (Or 196,882 in characteristic 2). problem Maximal It contains at least 20 of the 26 sporadic groups. subgroups This is one of the hardest simple groups to work with because there is no ”nice” way to look at it. For instance, Alt(100) is of much larger order, but because we can define this as permutations on 100 points it is much easier to deal with. Number Theory Group Theory

Classify all ‘primes’ HARD HARD BUT DONE

Construct elements from their EASY HARD AND NOT DONE ‘prime factors’

Numbers have unique factorisation, so it is very easy to construct the unique number with a given factorisation. For groups this isn’t anything like as straightforward.

The problem with these factors

Finite Simple Groups

Daniel Rogers As discussed, the simple groups act for group theory like prime Why do we care about simple numbers do for number theory. However there is a problem: groups?

What do we know about simple groups?

What questions are there about groups in light of the classifcation? The extension problem Maximal subgroups HARD BUT DONE

Construct elements from their EASY HARD AND NOT DONE ‘prime factors’

Numbers have unique factorisation, so it is very easy to construct the unique number with a given factorisation. For groups this isn’t anything like as straightforward.

The problem with these factors

Finite Simple Groups

Daniel Rogers As discussed, the simple groups act for group theory like prime Why do we care about simple numbers do for number theory. However there is a problem: groups?

What do we know about Number Theory Group Theory simple groups?

What questions are there about Classify all ‘primes’ HARD groups in light of the classifcation? The extension problem Maximal subgroups Construct elements from their EASY HARD AND NOT DONE ‘prime factors’

Numbers have unique factorisation, so it is very easy to construct the unique number with a given factorisation. For groups this isn’t anything like as straightforward.

The problem with these factors

Finite Simple Groups

Daniel Rogers As discussed, the simple groups act for group theory like prime Why do we care about simple numbers do for number theory. However there is a problem: groups?

What do we know about Number Theory Group Theory simple groups?

What questions are there about Classify all ‘primes’ HARD HARD BUT DONE groups in light of the classifcation? The extension problem Maximal subgroups EASY HARD AND NOT DONE

Numbers have unique factorisation, so it is very easy to construct the unique number with a given factorisation. For groups this isn’t anything like as straightforward.

The problem with these factors

Finite Simple Groups

Daniel Rogers As discussed, the simple groups act for group theory like prime Why do we care about simple numbers do for number theory. However there is a problem: groups?

What do we know about Number Theory Group Theory simple groups?

What questions are there about Classify all ‘primes’ HARD HARD BUT DONE groups in light of the classifcation? The extension problem Construct elements Maximal subgroups from their ‘prime factors’ HARD AND NOT DONE

Numbers have unique factorisation, so it is very easy to construct the unique number with a given factorisation. For groups this isn’t anything like as straightforward.

The problem with these factors

Finite Simple Groups

Daniel Rogers As discussed, the simple groups act for group theory like prime Why do we care about simple numbers do for number theory. However there is a problem: groups?

What do we know about Number Theory Group Theory simple groups?

Finite Simple Groups

Daniel Rogers As discussed, the simple groups act for group theory like prime Why do we care about simple numbers do for number theory. However there is a problem: groups?

What do we know about Number Theory Group Theory simple groups?

Numbers have unique factorisation, so it is very easy to construct the unique number with a given factorisation. For groups this isn’t anything like as straightforward. 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 the sporadic simple group M11.

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

Example - Alt(6)

Finite Simple Groups

Daniel Rogers

Why do we care about simple Example groups?

What do we Let G = h(1, 2)(3, 4)(5, 6)(7, 8), (2, 4, 8, 9)(3, 7, 6, 0)i. In fact know about ∼ simple groups? G = Alt(6), and so |G| = 360. There are various ways to add an

What questions element to G to give us a group of order 720. are there about groups in light of the classifcation? The extension problem Maximal subgroups 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 the sporadic simple group M11.

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

Example - Alt(6)

Finite Simple Groups

Daniel Rogers

Why do we care about simple Example groups?

What do we Let G = h(1, 2)(3, 4)(5, 6)(7, 8), (2, 4, 8, 9)(3, 7, 6, 0)i. In fact know about ∼ simple groups? G = Alt(6), and so |G| = 360. There are various ways to add an

What questions element to G to give us a group of order 720. are there about groups in light of Adding s := (1, 3, 7, 4, 2, 8)(5, 0, 9) gives Sym(6). the classifcation? 1 The extension problem Maximal subgroups Adding s1s2 gives M10, the stabiliser of a single point in the sporadic simple group M11.

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

Example - Alt(6)

Finite Simple Groups

Daniel Rogers

Why do we care about simple Example groups?

What do we Let G = h(1, 2)(3, 4)(5, 6)(7, 8), (2, 4, 8, 9)(3, 7, 6, 0)i. In fact know about ∼ simple groups? G = Alt(6), and so |G| = 360. There are various ways to add an

What questions element to G to give us a group of order 720. are there about groups in light of Adding s := (1, 3, 7, 4, 2, 8)(5, 0, 9) gives Sym(6). the classifcation? 1 The extension problem Adding s2 := (1, 8, 4, 6, 2, 7, 3, 5) gives PGL(2, 9). Maximal subgroups These all have the same composition factors, namely {Alt(6), C2}, but none of these three groups are isomorphic.

Example - Alt(6)

Finite Simple Groups

Daniel Rogers

Why do we care about simple Example groups?

What do we Let G = h(1, 2)(3, 4)(5, 6)(7, 8), (2, 4, 8, 9)(3, 7, 6, 0)i. In fact know about ∼ simple groups? G = Alt(6), and so |G| = 360. There are various ways to add an

What questions element to G to give us a group of order 720. are there about groups in light of Adding s := (1, 3, 7, 4, 2, 8)(5, 0, 9) gives Sym(6). the classifcation? 1 The extension problem Adding s2 := (1, 8, 4, 6, 2, 7, 3, 5) gives PGL(2, 9). Maximal subgroups Adding s1s2 gives M10, the stabiliser of a single point in the sporadic simple group M11. Example - Alt(6)

Finite Simple Groups

Daniel Rogers

Why do we care about simple Example groups?

What do we Let G = h(1, 2)(3, 4)(5, 6)(7, 8), (2, 4, 8, 9)(3, 7, 6, 0)i. In fact know about ∼ simple groups? G = Alt(6), and so |G| = 360. There are various ways to add an

What questions element to G to give us a group of order 720. are there about groups in light of Adding s := (1, 3, 7, 4, 2, 8)(5, 0, 9) gives Sym(6). the classifcation? 1 The extension problem Adding s2 := (1, 8, 4, 6, 2, 7, 3, 5) gives PGL(2, 9). Maximal subgroups Adding s1s2 gives M10, the stabiliser of a single point in the sporadic simple group M11.

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

Finite Simple Groups

Daniel Rogers

Why do we care about simple groups?

What do we know about simple groups? Given two abstract groups H and N, we can ask for a classification of What questions ∼ are there about all groups G such that G has a normal subgroup N and G/N = H. groups in light of the classifcation? Here G is called an extension of H by N, and finding this The extension problem classification is known as the extension problem. Maximal subgroups A semidirect product (sometimes called a split extension) Alt(6) o C2. All three of the groups described earlier fall into this category (and these are the only such ones in this case). A central extension - this is a group G such that H is contained in the center of G, and G/H =∼ K. There is a perfect central . extension of Alt(6) by C2, usually denoted 2 Alt(6). Here this is SL(2, 9). This is all possible groups with this structure.

Types of extension

Finite Simple Groups

Daniel Rogers

Why do we care about simple Some groups with composition factors {Alt(6), C2} are: groups?

What do we The direct product Alt(6) × C2. know about simple groups?

What questions are there about groups in light of the classifcation? The extension problem Maximal subgroups A central extension - this is a group G such that H is contained in the center of G, and G/H =∼ K. There is a perfect central . extension of Alt(6) by C2, usually denoted 2 Alt(6). Here this is SL(2, 9). This is all possible groups with this structure.

Types of extension

Finite Simple Groups

Daniel Rogers

Why do we care about simple Some groups with composition factors {Alt(6), C2} are: groups?

Types of extension

Finite Simple Groups

Daniel Rogers

Why do we care about simple Some groups with composition factors {Alt(6), C2} are: groups?

What do we The direct product Alt(6) × C2. know about simple groups? A semidirect product (sometimes called a split extension) What questions Alt(6) C2. All three of the groups described earlier fall into are there about o groups in light of this category (and these are the only such ones in this case). the classifcation? The extension problem A central extension - this is a group G such that H is contained Maximal ∼ subgroups in the center of G, and G/H = K. There is a perfect central . extension of Alt(6) by C2, usually denoted 2 Alt(6). Here this is SL(2, 9). Types of extension

Finite Simple Groups

Daniel Rogers

Why do we care about simple Some groups with composition factors {Alt(6), C2} are: groups?

Finite Simple Groups

Daniel Rogers

Why do we care about simple groups?

What do we know about simple groups?

What questions The extension problem is hard and currently unsolved! Understanding are there about groups in light of this problem would allow us to produce the Classiﬁcation of Finite the classifcation? Groups. The extension problem Maximal subgroups Example Let G = Alt(5) = h(1, 2)(3, 4), (1, 2, 3, 4, 5)i of order 60. Then its maximal subgroups (up to conjugacy) are: Alt(4) =∼ h(1, 2)(3, 4), (1, 2, 3)i, order 12. ∼ D5 = h(1, 2, 3, 4, 5), (1, 5)(2, 4)i, order 10. Sym(3) =∼ h(1, 2, 3), (2, 3)(4, 5)i, order 6.

Maximal Subgroups

Finite Simple Groups

Daniel Rogers

Why do we care Deﬁnition about simple groups? A subgroup H < G is maximal if H 6= G and there is no subgroup M What do we know about such that H M G. simple groups?

What questions are there about groups in light of the classifcation? The extension problem Maximal subgroups Maximal Subgroups

Finite Simple Groups

Daniel Rogers

Why do we care Deﬁnition about simple groups? A subgroup H < G is maximal if H 6= G and there is no subgroup M What do we know about such that H M G. simple groups?

What questions are there about groups in light of Example the classifcation? The extension Let G = Alt(5) = h(1, 2)(3, 4), (1, 2, 3, 4, 5)i of order 60. Then its problem Maximal maximal subgroups (up to conjugacy) are: subgroups Alt(4) =∼ h(1, 2)(3, 4), (1, 2, 3)i, order 12. ∼ D5 = h(1, 2, 3, 4, 5), (1, 5)(2, 4)i, order 10. Sym(3) =∼ h(1, 2, 3), (2, 3)(4, 5)i, order 6. Example Alt(5) has no subgroup of order 15.

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

Why are maximal subgroups useful?

Finite Simple Groups

Daniel Rogers

Why do we care about simple groups? Understanding maximal subgroups allows us to understand all What do we subgroups. know about simple groups?

What questions are there about groups in light of the classifcation? The extension problem Maximal subgroups There is a 1:1 correspondence between maximal subgroups of a simple group, and primitive permutation groups isomorphic to them. So understanding maximal subgroups allows us to write a given group in various diﬀerent ways as permutation groups.

Why are maximal subgroups useful?

Finite Simple Groups

Daniel Rogers

Why do we care about simple groups? Understanding maximal subgroups allows us to understand all What do we subgroups. know about simple groups?

What questions Example are there about groups in light of Alt(5) has no subgroup of order 15. the classifcation? The extension problem Maximal subgroups Why are maximal subgroups useful?

Finite Simple Groups

Daniel Rogers

Why do we care about simple groups? Understanding maximal subgroups allows us to understand all What do we subgroups. know about simple groups?

What questions Example are there about groups in light of Alt(5) has no subgroup of order 15. the classifcation? The extension problem Maximal There is a 1:1 correspondence between maximal subgroups of a subgroups simple group, and primitive permutation groups isomorphic to them. So understanding maximal subgroups allows us to write a given group in various diﬀerent ways as permutation groups. Why are maximal subgroups useful?

Finite Simple Groups

Daniel Rogers

Why do we care Example about simple groups? The maximal subgroups of Alt(5) (a group of order 60) are Alt(4) What do we know about (order 12), D5 (order 10) and Sym(3) (order 6). These correspond to simple groups? three different permutation representations of Alt(5): What questions are there about groups in light of Alt(4) gives us h(1, 2, 3, 4, 5), (1, 2)(3, 4)i. the classifcation? The extension D5 gives us h(1, 3, 4)(2, 5, 6), (2, 6, 3, 5, 4)i. problem Maximal subgroups Sym(3) gives us h(1, 2, 3, 4, 5)(6, 7, 8, 9, 0), (1, 2)(5, 0)(4, 7)(3, 9)i. These are all the ”genuinely different” permutation representations of Alt(5) (without being precise as to what I mean by ”genuinely different”) The O’Nan-Scott theorem

Finite Simple Groups

Daniel Rogers Discovered independently by O’Nan and Scott, this classiﬁes the Why do we care about simple maximal subgroups of the symmetric group. groups? What do we Theorem know about simple groups? The maximal subgroups of Sym(n) is contained in (at least one of): What questions are there about Sk × Sn−k for some k, 0 < k < n. groups in light of the classifcation? The extension A wreath product Sk o Sl where kl = n. problem Maximal A selection of speciﬁc primitive groups subgroups

Proof. Relatively unpleasant (but not too long). See for example The Finite Simple Groups by Wilson. Aschbacher’s Theorem

Finite Simple Groups

Daniel Rogers

Why do we care about simple groups? Aschbacher gives a classification of maximal subgroups of classical What do we groups (recall earlier) into 9 classes, denoted C1, ..., C9, although know about simple groups? these classes can’t be described in a slide very easily. What questions are there about Classes C1, ..., C8 have geometric structure, and these classes groups in light of the classifcation? have been classified completely by Kleidman and Liebeck. The extension problem C9 is essentially ”everything else” and requires more case by case Maximal subgroups analysis. In particular these generally have to be studied in individual dimensions and it is very unlikely that a full classification of all the groups in this class will be possible. Class C9

Finite Simple Groups

Daniel Rogers

Why do we care about simple groups?

What do we know about simple groups? A lot of the groups that occur in this class relate to the irreducible What questions representations of various simple groups. If a group has an irreducible are there about groups in light of character of degree n then this group will exist as a subgroup of the classifcation? e The extension SL(n, q) where q = p (and possibly inside another classical group) problem Maximal and, with some technicalities, these are often maximal. subgroups Classifying C9 subgroups

Finite Simple Groups

Daniel Rogers

Why do we care about simple groups?

What do we know about simple groups? Dimensions up to 12 done by Bray, Holt and Roney-Dougal What questions are there about (2013). groups in light of the classifcation? Dimensions 13, 14 and 15 work in progress by Schroeder. The extension problem Dimensions 16 and 17 work in progress by R . Maximal subgroups