The Finite Simple Groups I: Description

Nick Gill (OU) The Finite Simple Groups I: Description

Nick Gill (OU)

November 14, 2012 Galois introduced the notion of a simple group and observed that Alt(5) was simple.

In this talk we will be interested in the finite simple groups. The study of infinite simple groups is an entirely different proposition: much wilder and much less understood.

Simple groups

The Finite Simple Groups I: Description Deﬁnition Nick Gill (OU) A group G is called simple if it has no non-trivial proper normal subgroups. Galois introduced the notion of a simple group and observed that Alt(5) was simple.

In this talk we will be interested in the finite simple groups. The study of infinite simple groups is an entirely different proposition: much wilder and much less understood.

Simple groups

The Finite Simple Groups I: Description Definition Nick Gill (OU) A group G is called simple if it has no non-trivial proper normal subgroups. In this talk we will be interested in the finite simple groups. The study of infinite simple groups is an entirely different proposition: much wilder and much less understood.

Simple groups

The Finite Simple Groups I: Description Deﬁnition Nick Gill (OU) A group G is called simple if it has no non-trivial proper normal subgroups.

Galois introduced the notion of a simple group and observed that Alt(5) was simple. Simple groups

The Finite Simple Groups I: Description Deﬁnition Nick Gill (OU) A group G is called simple if it has no non-trivial proper normal subgroups.

Galois introduced the notion of a simple group and observed that Alt(5) was simple.

In this talk we will be interested in the finite simple groups. The study of infinite simple groups is an entirely different proposition: much wilder and much less understood. In 1981 CFSG was announced, although it was only in 2001 that all parts of the proof were written up and published.

John Thompson Michael Aschbacher

The Classiﬁcation of Finite Simple Groups

The Finite Simple Groups I: Description

Nick Gill (OU) Michael Aschbacher

The Classiﬁcation of Finite Simple Groups

The Finite Simple Groups In 1981 CFSG was announced, although it was only in 2001 I: Description that all parts of the proof were written up and published. Nick Gill (OU)

John Thompson The Classiﬁcation of Finite Simple Groups

The Finite Simple Groups In 1981 CFSG was announced, although it was only in 2001 I: Description that all parts of the proof were written up and published. Nick Gill (OU)

John Thompson Michael Aschbacher 1 the cyclic groups of prime order; 2 the alternating groups of degree at least 5; 3 the ﬁnite groups of Lie type; 4 the 26 sporadic groups.

Theorem Let G be a ﬁnite simple group. Then G lies in one (or more) of the following families:

The four families are pair-wise disjoint except for (2) and (3). We will spend the rest of the lecture discussing their various properties.

The Classiﬁcation of Finite Simple Groups

The Finite Simple Groups I: Description

Nick Gill (OU) 1 the cyclic groups of prime order; 2 the alternating groups of degree at least 5; 3 the ﬁnite groups of Lie type; 4 the 26 sporadic groups.

The four families are pair-wise disjoint except for (2) and (3). We will spend the rest of the lecture discussing their various properties.

The Classiﬁcation of Finite Simple Groups

The Finite Simple Groups I: Description Nick Gill Theorem (OU) Let G be a ﬁnite simple group. Then G lies in one (or more) of the following families: 2 the alternating groups of degree at least 5; 3 the ﬁnite groups of Lie type; 4 the 26 sporadic groups.

The four families are pair-wise disjoint except for (2) and (3). We will spend the rest of the lecture discussing their various properties.

The Classiﬁcation of Finite Simple Groups

The four families are pair-wise disjoint except for (2) and (3). We will spend the rest of the lecture discussing their various properties.

The Classiﬁcation of Finite Simple Groups

The four families are pair-wise disjoint except for (2) and (3). We will spend the rest of the lecture discussing their various properties.

The Classiﬁcation of Finite Simple Groups

The Finite Simple Groups I: Description Nick Gill Theorem (OU) Let G be a ﬁnite simple group. Then G lies in one (or more) of the following families: 1 the cyclic groups of prime order; 2 the alternating groups of degree at least 5; 3 the ﬁnite groups of Lie type; The four families are pair-wise disjoint except for (2) and (3). We will spend the rest of the lecture discussing their various properties.

The Classiﬁcation of Finite Simple Groups

The Finite Simple Groups I: Description Nick Gill Theorem (OU) Let G be a finite simple group. Then G lies in one (or more) of the following families: 1 the cyclic groups of prime order; 2 the alternating groups of degree at least 5; 3 the finite groups of Lie type; 4 the 26 sporadic groups. The Classification of Finite Simple Groups

The four families are pair-wise disjoint except for (2) and (3). We will spend the rest of the lecture discussing their various properties. Theorem ∼ G is a ﬁnite abelian simple group if and only if G = Cp, a cyclic group of prime order.

The cyclic groups of prime order

The Finite Simple Groups I: Description

Nick Gill (OU) The cyclic groups of prime order

The Finite Simple Groups I: Description

Nick Gill (OU)

Theorem ∼ G is a ﬁnite abelian simple group if and only if G = Cp, a cyclic group of prime order. The natural habitat of a group is inside Sym(Ω) for some set Ω, i.e. all groups are naturally permutation groups. Suppose that G ≤ Sym(Ω). We say that G is transitive if ∀α, β ∈ Ω, ∃g ∈ G, αg = β.

We say that G is k-transitive if

∗k ∀(α1, . . . , αk ), (β1, . . . , βk ) ∈ Ω ,

∃g ∈ G, (α1g, α2g, . . . , αk g) = (β1, . . . , βk ).

k-transitive groups have a tendency towards simplicity... Theorem (Burnside) A 2-transitive group G has a unique minimal normal subgroup, which is either elementary-abelian or simple.

Multiply-transitive simple groups

The Finite Simple Groups I: Description

Nick Gill (OU) Suppose that G ≤ Sym(Ω). We say that G is transitive if ∀α, β ∈ Ω, ∃g ∈ G, αg = β.

We say that G is k-transitive if

∗k ∀(α1, . . . , αk ), (β1, . . . , βk ) ∈ Ω ,

∃g ∈ G, (α1g, α2g, . . . , αk g) = (β1, . . . , βk ).

k-transitive groups have a tendency towards simplicity... Theorem (Burnside) A 2-transitive group G has a unique minimal normal subgroup, which is either elementary-abelian or simple.

Multiply-transitive simple groups

The Finite The natural habitat of a group is inside Sym(Ω) for some set Simple Groups I: Description Ω, i.e. all groups are naturally permutation groups. Nick Gill (OU) We say that G is k-transitive if

∗k ∀(α1, . . . , αk ), (β1, . . . , βk ) ∈ Ω ,

∃g ∈ G, (α1g, α2g, . . . , αk g) = (β1, . . . , βk ).

k-transitive groups have a tendency towards simplicity... Theorem (Burnside) A 2-transitive group G has a unique minimal normal subgroup, which is either elementary-abelian or simple.

Multiply-transitive simple groups

The Finite The natural habitat of a group is inside Sym(Ω) for some set Simple Groups I: Description Ω, i.e. all groups are naturally permutation groups. Nick Gill (OU) Suppose that G ≤ Sym(Ω). We say that G is transitive if ∀α, β ∈ Ω, ∃g ∈ G, αg = β. k-transitive groups have a tendency towards simplicity... Theorem (Burnside) A 2-transitive group G has a unique minimal normal subgroup, which is either elementary-abelian or simple.

Multiply-transitive simple groups

We say that G is k-transitive if

∗k ∀(α1, . . . , αk ), (β1, . . . , βk ) ∈ Ω ,

∃g ∈ G, (α1g, α2g, . . . , αk g) = (β1, . . . , βk ). Multiply-transitive simple groups

We say that G is k-transitive if

∗k ∀(α1, . . . , αk ), (β1, . . . , βk ) ∈ Ω ,

∃g ∈ G, (α1g, α2g, . . . , αk g) = (β1, . . . , βk ).

k-transitive groups have a tendency towards simplicity... Theorem (Burnside) A 2-transitive group G has a unique minimal normal subgroup, which is either elementary-abelian or simple.

Suppose that N is a non-trivial proper normal subgroup of G. 1 The conjugacy class of 3-cycles generate G. 2 If N contains a 3-cycle, then .

3 If g is non-trivial in N, then there are conjugates g1,..., gl of g such that g1 ··· gl is a 3-cycle.

Alternating groups

The Finite Simple Groups I: Description Nick Gill Theorem (OU) When n ≥ 5,G = Alt(n) is simple.

Proof.

1 The conjugacy class of 3-cycles generate G. 2 If N contains a 3-cycle, then .

3 If g is non-trivial in N, then there are conjugates g1,..., gl of g such that g1 ··· gl is a 3-cycle.

Alternating groups

The Finite Simple Groups I: Description Nick Gill Theorem (OU) When n ≥ 5,G = Alt(n) is simple.

Proof. Suppose that N is a non-trivial proper normal subgroup of G.

2 If N contains a 3-cycle, then .

3 If g is non-trivial in N, then there are conjugates g1,..., gl of g such that g1 ··· gl is a 3-cycle.

Alternating groups

The Finite Simple Groups I: Description Nick Gill Theorem (OU) When n ≥ 5,G = Alt(n) is simple.

Proof. Suppose that N is a non-trivial proper normal subgroup of G. 1 The conjugacy class of 3-cycles generate G.

3 If g is non-trivial in N, then there are conjugates g1,..., gl of g such that g1 ··· gl is a 3-cycle.

Alternating groups

The Finite Simple Groups I: Description Nick Gill Theorem (OU) When n ≥ 5,G = Alt(n) is simple.

Proof. Suppose that N is a non-trivial proper normal subgroup of G. 1 The conjugacy class of 3-cycles generate G. 2 If N contains a 3-cycle, then .

Alternating groups

The Finite Simple Groups I: Description Nick Gill Theorem (OU) When n ≥ 5,G = Alt(n) is simple.

Proof. Suppose that N is a non-trivial proper normal subgroup of G. 1 The conjugacy class of 3-cycles generate G. 2 If N contains a 3-cycle, then .

3 If g is non-trivial in N, then there are conjugates g1,..., gl of g such that g1 ··· gl is a 3-cycle. Alternating groups

The Finite Simple Groups I: Description Nick Gill Theorem (OU) When n ≥ 5,G = Alt(n) is simple.

Proof. Suppose that N is a non-trivial proper normal subgroup of G. 1 The conjugacy class of 3-cycles generate G. 2 If N contains a 3-cycle, then .

3 If g is non-trivial in N, then there are conjugates g1,..., gl of g such that g1 ··· gl is a 3-cycle.

Note that M12 (and M11 in its 4-transitive incarnation) are sharply k-transitive.

Theorem Other than Alt(n) and Sym(n), the ﬁnite almost simple 3-transitive groups are:

1 (5-trans):M24 and M12 (twice);

2 (4-trans):M23 and M11;

3 (3-trans):M22,M11 and PGL2(q) with q ≥ 4.

So far the only proof that these really are all the almost simple 3-transitive groups relies on CFSG.

The Mathieu groups I

The Finite Simple Groups I: Description The ﬁrst ﬁve sporadic groups are called the Mathieu groups. Nick Gill (OU) They arise ‘naturally’: Note that M12 (and M11 in its 4-transitive incarnation) are sharply k-transitive.

So far the only proof that these really are all the almost simple 3-transitive groups relies on CFSG.

The Mathieu groups I

The Finite Simple Groups I: Description The first five sporadic groups are called the Mathieu groups. Nick Gill (OU) They arise ‘naturally’: Theorem Other than Alt(n) and Sym(n), the finite almost simple 3-transitive groups are:

1 (5-trans):M24 and M12 (twice);

2 (4-trans):M23 and M11;

3 (3-trans):M22,M11 and PGL2(q) with q ≥ 4. Note that M12 (and M11 in its 4-transitive incarnation) are sharply k-transitive.

The Mathieu groups I

1 (5-trans):M24 and M12 (twice);

2 (4-trans):M23 and M11;

3 (3-trans):M22,M11 and PGL2(q) with q ≥ 4.

So far the only proof that these really are all the almost simple 3-transitive groups relies on CFSG. The Mathieu groups I

1 (5-trans):M24 and M12 (twice);

2 (4-trans):M23 and M11;

3 (3-trans):M22,M11 and PGL2(q) with q ≥ 4.

So far the only proof that these really are all the almost simple 3-transitive groups relies on CFSG. Note that M12 (and M11 in its 4-transitive incarnation) are sharply k-transitive. Emile Mathieu introduced M12 and M24 in an 1861 paper. Here’s some of the introduction.

The Mathieu groups II

The Finite Simple Groups I: Description

Nick Gill (OU) There was some uncertainty about Mathieu’s results so in 1873 he wrote another paper, from which I quote: ...if no expert was able to ﬁll in the details of my claims made twelve years ago, I’d better do it myself.

The Mathieu groups II

The Finite Simple Groups Emile Mathieu introduced M12 and M24 in I: Description an 1861 paper. Here’s some of the Nick Gill (OU) introduction. There was some uncertainty about Mathieu’s results so in 1873 he wrote another paper, from which I quote: ...if no expert was able to ﬁll in the details of my claims made twelve years ago, I’d better do it myself.

The Mathieu groups II

The Finite Simple Groups Emile Mathieu introduced M12 and M24 in I: Description an 1861 paper. Here’s some of the Nick Gill (OU) introduction. ...if no expert was able to ﬁll in the details of my claims made twelve years ago, I’d better do it myself.

The Mathieu groups II

The Finite Simple Groups Emile Mathieu introduced M12 and M24 in I: Description an 1861 paper. Here’s some of the Nick Gill (OU) introduction.

There was some uncertainty about Mathieu’s results so in 1873 he wrote another paper, from which I quote: The Mathieu groups II

The Finite Simple Groups Emile Mathieu introduced M12 and M24 in I: Description an 1861 paper. Here’s some of the Nick Gill (OU) introduction.

There was some uncertainty about Mathieu’s results so in 1873 he wrote another paper, from which I quote: ...if no expert was able to ﬁll in the details of my claims made twelve years ago, I’d better do it myself. Deﬁnition A Steiner System, S(k, m, n) is a set Ω of size n and a set of subsets of Ω, each of size m such that any set of k elements of Ω lies in exactly one of these subsets.

Theorem There exist unique SS S(5, 6, 12) and S(5, 8, 24) such that

Aut(S(5, 6, 12)) = M12 and Aut(S(5, 8, 24)) = M24.

Given a SS S(k, m, n) and an element α ∈ Ω, one obtains another SS S(k − 1, m − 1, n − 1) on Ω\{α} by restricting to the subset of S(k, m, n) containing the element α.

The Mathieu groups III

The Finite In 1938, Witt constructed the Mathieu groups geometrically, Simple Groups I: Description thereby laying all questions about their existence to rest. Nick Gill (OU) Theorem There exist unique SS S(5, 6, 12) and S(5, 8, 24) such that

Aut(S(5, 6, 12)) = M12 and Aut(S(5, 8, 24)) = M24.

Given a SS S(k, m, n) and an element α ∈ Ω, one obtains another SS S(k − 1, m − 1, n − 1) on Ω\{α} by restricting to the subset of S(k, m, n) containing the element α.

The Mathieu groups III

The Finite In 1938, Witt constructed the Mathieu groups geometrically, Simple Groups I: Description thereby laying all questions about their existence to rest. Nick Gill (OU) Deﬁnition A Steiner System, S(k, m, n) is a set Ω of size n and a set of subsets of Ω, each of size m such that any set of k elements of Ω lies in exactly one of these subsets. Given a SS S(k, m, n) and an element α ∈ Ω, one obtains another SS S(k − 1, m − 1, n − 1) on Ω\{α} by restricting to the subset of S(k, m, n) containing the element α.

The Mathieu groups III

Theorem There exist unique SS S(5, 6, 12) and S(5, 8, 24) such that

Aut(S(5, 6, 12)) = M12 and Aut(S(5, 8, 24)) = M24. The Mathieu groups III

Theorem There exist unique SS S(5, 6, 12) and S(5, 8, 24) such that

Aut(S(5, 6, 12)) = M12 and Aut(S(5, 8, 24)) = M24.

Given a SS S(k, m, n) and an element α ∈ Ω, one obtains another SS S(k − 1, m − 1, n − 1) on Ω\{α} by restricting to the subset of S(k, m, n) containing the element α. These are central quotients of matrix groups over a ﬁnite ﬁeld. The archetypal example is PSLn(q):

1 Deﬁne the group SLn(q) to be the set of all n × n matrices over a ﬁeld of order q with determinant 1.

2 Deﬁne Z to be the center of SLn(q). This is a group of scalar matrices of size (n, q − 1).

3 Deﬁne the group PSLn(q) = SLn(q)/Z.

Theorem

If (n, q) 6∈ {(2, 2), (2, 3)}, then PSLn(q) is simple.

This is proved using Iwasawa’s criterion and the fact that the natural action of PSLn(q) on 1-spaces is 2-transitive.

Simple groups of Lie type

The Finite Simple Groups I: Description

Nick Gill (OU) 1 Deﬁne the group SLn(q) to be the set of all n × n matrices over a ﬁeld of order q with determinant 1.

2 Deﬁne Z to be the center of SLn(q). This is a group of scalar matrices of size (n, q − 1).

3 Deﬁne the group PSLn(q) = SLn(q)/Z.

Theorem

If (n, q) 6∈ {(2, 2), (2, 3)}, then PSLn(q) is simple.

This is proved using Iwasawa’s criterion and the fact that the natural action of PSLn(q) on 1-spaces is 2-transitive.

Simple groups of Lie type

The Finite Simple Groups I: Description These are central quotients of matrix groups over a finite field. Nick Gill (OU) The archetypal example is PSLn(q): 2 Define Z to be the center of SLn(q). This is a group of scalar matrices of size (n, q − 1).

3 Deﬁne the group PSLn(q) = SLn(q)/Z.

Theorem

If (n, q) 6∈ {(2, 2), (2, 3)}, then PSLn(q) is simple.

This is proved using Iwasawa’s criterion and the fact that the natural action of PSLn(q) on 1-spaces is 2-transitive.

Simple groups of Lie type

The Finite Simple Groups I: Description These are central quotients of matrix groups over a finite field. Nick Gill (OU) The archetypal example is PSLn(q): 1 Define the group SLn(q) to be the set of all n × n matrices over a field of order q with determinant 1. 3 Define the group PSLn(q) = SLn(q)/Z.

Theorem

If (n, q) 6∈ {(2, 2), (2, 3)}, then PSLn(q) is simple.

This is proved using Iwasawa’s criterion and the fact that the natural action of PSLn(q) on 1-spaces is 2-transitive.

Simple groups of Lie type

2 Deﬁne Z to be the center of SLn(q). This is a group of scalar matrices of size (n, q − 1). Theorem

If (n, q) 6∈ {(2, 2), (2, 3)}, then PSLn(q) is simple.

This is proved using Iwasawa’s criterion and the fact that the natural action of PSLn(q) on 1-spaces is 2-transitive.

Simple groups of Lie type

2 Deﬁne Z to be the center of SLn(q). This is a group of scalar matrices of size (n, q − 1).

3 Deﬁne the group PSLn(q) = SLn(q)/Z. This is proved using Iwasawa’s criterion and the fact that the natural action of PSLn(q) on 1-spaces is 2-transitive.

Simple groups of Lie type

2 Deﬁne Z to be the center of SLn(q). This is a group of scalar matrices of size (n, q − 1).

3 Deﬁne the group PSLn(q) = SLn(q)/Z.

Theorem

If (n, q) 6∈ {(2, 2), (2, 3)}, then PSLn(q) is simple. Simple groups of Lie type

2 Deﬁne Z to be the center of SLn(q). This is a group of scalar matrices of size (n, q − 1).

3 Deﬁne the group PSLn(q) = SLn(q)/Z.

Theorem

If (n, q) 6∈ {(2, 2), (2, 3)}, then PSLn(q) is simple.

This is proved using Iwasawa’s criterion and the fact that the natural action of PSLn(q) on 1-spaces is 2-transitive. Camille Jordan Leonard Dickson

The rest of the classical groups

The Finite Simple Groups I: Description

Nick Gill (OU) Leonard Dickson

The rest of the classical groups

The Finite Simple Groups I: Description

Nick Gill (OU)

Camille Jordan The rest of the classical groups

The Finite Simple Groups I: Description

Nick Gill (OU)

Camille Jordan Leonard Dickson The ﬁnite classical groups have been known for a long time. We construct them using forms on a vector space.

1 Let V be a vector space of dimension n over Fq, a ﬁeld of order q.

2 Let φ : V × V → Fq be a bilinear form. 3 Let Gφ < GLn(q) be the group of isometries of φ, i.e.

Gφ = {g ∈ GLn(q) | φ(gv, gw) = φ(v, w)∀v, w ∈ V }.

The group Gφ, as well as some of its subgroups and their quotients, is known as a ﬁnite classical group.

The rest of the classical groups

The Finite Simple Groups I: Description

Nick Gill (OU) 1 Let V be a vector space of dimension n over Fq, a ﬁeld of order q.

2 Let φ : V × V → Fq be a bilinear form. 3 Let Gφ < GLn(q) be the group of isometries of φ, i.e.

Gφ = {g ∈ GLn(q) | φ(gv, gw) = φ(v, w)∀v, w ∈ V }.

The group Gφ, as well as some of its subgroups and their quotients, is known as a ﬁnite classical group.

The rest of the classical groups

The Finite Simple Groups I: Description The ﬁnite classical groups have been known for a long time. Nick Gill (OU) We construct them using forms on a vector space. 2 Let φ : V × V → Fq be a bilinear form. 3 Let Gφ < GLn(q) be the group of isometries of φ, i.e.

Gφ = {g ∈ GLn(q) | φ(gv, gw) = φ(v, w)∀v, w ∈ V }.

The group Gφ, as well as some of its subgroups and their quotients, is known as a ﬁnite classical group.

The rest of the classical groups

The Finite Simple Groups I: Description The ﬁnite classical groups have been known for a long time. Nick Gill (OU) We construct them using forms on a vector space.

1 Let V be a vector space of dimension n over Fq, a ﬁeld of order q. 3 Let Gφ < GLn(q) be the group of isometries of φ, i.e.

Gφ = {g ∈ GLn(q) | φ(gv, gw) = φ(v, w)∀v, w ∈ V }.

The group Gφ, as well as some of its subgroups and their quotients, is known as a ﬁnite classical group.

The rest of the classical groups

The Finite Simple Groups I: Description The ﬁnite classical groups have been known for a long time. Nick Gill (OU) We construct them using forms on a vector space.

1 Let V be a vector space of dimension n over Fq, a ﬁeld of order q.

2 Let φ : V × V → Fq be a bilinear form. The group Gφ, as well as some of its subgroups and their quotients, is known as a ﬁnite classical group.

The rest of the classical groups

The Finite Simple Groups I: Description The ﬁnite classical groups have been known for a long time. Nick Gill (OU) We construct them using forms on a vector space.

1 Let V be a vector space of dimension n over Fq, a ﬁeld of order q.

2 Let φ : V × V → Fq be a bilinear form. 3 Let Gφ < GLn(q) be the group of isometries of φ, i.e.

Gφ = {g ∈ GLn(q) | φ(gv, gw) = φ(v, w)∀v, w ∈ V }. The rest of the classical groups

The Finite Simple Groups I: Description The ﬁnite classical groups have been known for a long time. Nick Gill (OU) We construct them using forms on a vector space.

1 Let V be a vector space of dimension n over Fq, a ﬁeld of order q.

2 Let φ : V × V → Fq be a bilinear form. 3 Let Gφ < GLn(q) be the group of isometries of φ, i.e.

Gφ = {g ∈ GLn(q) | φ(gv, gw) = φ(v, w)∀v, w ∈ V }.

The group Gφ, as well as some of its subgroups and their quotients, is known as a ﬁnite classical group. If φ is alternating, then Gφ = Spn(q), a symplectic group. Theorem

If n ≥ 4, then PSpn(q) = Spn(q)/Z(Spn(q)) is simple.

If φ is symmetric and q is odd, then Gφ = GOn(q), an orthogonal group. Theorem If n ≥ 5, then GOn(q) has a normal subgroup Ωn(q) such that PΩn(q) = Ωn(q)/Z(Ωn(q)) is simple.

The remaining classical groups – PSUn(q)(unitary) and a PΩn(2 )(orthogonal with q even) – are obtained by studying isometries of sesquilinear and quadratic forms.

Symplectic groups and (some) orthogonal groups

The Finite Simple Groups The theory of bilinear forms tells us that, if φ is I: Description non-degenerate, then it is either alternating or symmetric. Nick Gill (OU) Theorem

If n ≥ 4, then PSpn(q) = Spn(q)/Z(Spn(q)) is simple.

If φ is symmetric and q is odd, then Gφ = GOn(q), an orthogonal group. Theorem If n ≥ 5, then GOn(q) has a normal subgroup Ωn(q) such that PΩn(q) = Ωn(q)/Z(Ωn(q)) is simple.

The remaining classical groups – PSUn(q)(unitary) and a PΩn(2 )(orthogonal with q even) – are obtained by studying isometries of sesquilinear and quadratic forms.

Symplectic groups and (some) orthogonal groups

The Finite Simple Groups The theory of bilinear forms tells us that, if φ is I: Description non-degenerate, then it is either alternating or symmetric. Nick Gill (OU) If φ is alternating, then Gφ = Spn(q), a symplectic group. If φ is symmetric and q is odd, then Gφ = GOn(q), an orthogonal group. Theorem If n ≥ 5, then GOn(q) has a normal subgroup Ωn(q) such that PΩn(q) = Ωn(q)/Z(Ωn(q)) is simple.

The remaining classical groups – PSUn(q)(unitary) and a PΩn(2 )(orthogonal with q even) – are obtained by studying isometries of sesquilinear and quadratic forms.

Symplectic groups and (some) orthogonal groups

If n ≥ 4, then PSpn(q) = Spn(q)/Z(Spn(q)) is simple. Theorem If n ≥ 5, then GOn(q) has a normal subgroup Ωn(q) such that PΩn(q) = Ωn(q)/Z(Ωn(q)) is simple.

The remaining classical groups – PSUn(q)(unitary) and a PΩn(2 )(orthogonal with q even) – are obtained by studying isometries of sesquilinear and quadratic forms.

Symplectic groups and (some) orthogonal groups

If n ≥ 4, then PSpn(q) = Spn(q)/Z(Spn(q)) is simple.

If φ is symmetric and q is odd, then Gφ = GOn(q), an orthogonal group. The remaining classical groups – PSUn(q)(unitary) and a PΩn(2 )(orthogonal with q even) – are obtained by studying isometries of sesquilinear and quadratic forms.

Symplectic groups and (some) orthogonal groups

If n ≥ 4, then PSpn(q) = Spn(q)/Z(Spn(q)) is simple.

If φ is symmetric and q is odd, then Gφ = GOn(q), an orthogonal group. Theorem If n ≥ 5, then GOn(q) has a normal subgroup Ωn(q) such that PΩn(q) = Ωn(q)/Z(Ωn(q)) is simple. Symplectic groups and (some) orthogonal groups

If n ≥ 4, then PSpn(q) = Spn(q)/Z(Spn(q)) is simple.

If φ is symmetric and q is odd, then Gφ = GOn(q), an orthogonal group. Theorem If n ≥ 5, then GOn(q) has a normal subgroup Ωn(q) such that PΩn(q) = Ωn(q)/Z(Ωn(q)) is simple.

The remaining classical groups – PSUn(q)(unitary) and a PΩn(2 )(orthogonal with q even) – are obtained by studying isometries of sesquilinear and quadratic forms. Theorem The simple complex Lie groups are SLn(C), Spn(C), SOn(C), G2(C), F4(C), E6(C), E7(C), E8(C). These groups are classiﬁed using Dynkin diagrams.

Connection to Lie groups

The Finite Simple Groups I: Description

Nick Gill The ﬁnite classical groups have (OU) big sisters amongst the simple Lie groups over C. These groups are classiﬁed using Dynkin diagrams.

Connection to Lie groups

The Finite Simple Groups I: Description

Nick Gill The ﬁnite classical groups have (OU) big sisters amongst the simple Lie groups over C. Theorem The simple complex Lie groups are SLn(C), Spn(C), SOn(C), G2(C), F4(C), E6(C), E7(C), E8(C). Connection to Lie groups

The Finite Simple Groups I: Description

Nick Gill (OU)

Claude Chevalley 1 Let G be a simple algebraic subgroup of GLn(Fq). 2 The possibilities for G are given by the Dynkin diagrams. F 3 Look at G = G , the subset of G fixed by the Frobenius q automorphism (aij ) 7→ (aij ). 4 It is easy to prove that G is a finite group. These groups are called the Chevalley groups. They include most of the finite classical groups, plus some more - the first exceptional groups:

G2(q), F4(q), E6(q), E7(q), E8(q).

Chevalley groups

The Finite Simple Groups In 1955 Claude Chevalley found a way to construct (some of) I: Description the finite classical groups using the Lie theory, rather than the Nick Gill (OU) theory of forms. 2 The possibilities for G are given by the Dynkin diagrams. F 3 Look at G = G , the subset of G fixed by the Frobenius q automorphism (aij ) 7→ (aij ). 4 It is easy to prove that G is a finite group. These groups are called the Chevalley groups. They include most of the finite classical groups, plus some more - the first exceptional groups:

G2(q), F4(q), E6(q), E7(q), E8(q).

Chevalley groups

The Finite Simple Groups In 1955 Claude Chevalley found a way to construct (some of) I: Description the finite classical groups using the Lie theory, rather than the Nick Gill (OU) theory of forms. 1 Let G be a simple algebraic subgroup of GLn(Fq). F 3 Look at G = G , the subset of G fixed by the Frobenius q automorphism (aij ) 7→ (aij ). 4 It is easy to prove that G is a finite group. These groups are called the Chevalley groups. They include most of the finite classical groups, plus some more - the first exceptional groups:

G2(q), F4(q), E6(q), E7(q), E8(q).

Chevalley groups

The Finite Simple Groups In 1955 Claude Chevalley found a way to construct (some of) I: Description the finite classical groups using the Lie theory, rather than the Nick Gill (OU) theory of forms. 1 Let G be a simple algebraic subgroup of GLn(Fq). 2 The possibilities for G are given by the Dynkin diagrams. 4 It is easy to prove that G is a finite group. These groups are called the Chevalley groups. They include most of the finite classical groups, plus some more - the first exceptional groups:

G2(q), F4(q), E6(q), E7(q), E8(q).

Chevalley groups

The Finite Simple Groups In 1955 Claude Chevalley found a way to construct (some of) I: Description the finite classical groups using the Lie theory, rather than the Nick Gill (OU) theory of forms. 1 Let G be a simple algebraic subgroup of GLn(Fq). 2 The possibilities for G are given by the Dynkin diagrams. F 3 Look at G = G , the subset of G fixed by the Frobenius q automorphism (aij ) 7→ (aij ). These groups are called the Chevalley groups. They include most of the finite classical groups, plus some more - the first exceptional groups:

G2(q), F4(q), E6(q), E7(q), E8(q).

Chevalley groups

The Finite Simple Groups In 1955 Claude Chevalley found a way to construct (some of) I: Description the finite classical groups using the Lie theory, rather than the Nick Gill (OU) theory of forms. 1 Let G be a simple algebraic subgroup of GLn(Fq). 2 The possibilities for G are given by the Dynkin diagrams. F 3 Look at G = G , the subset of G fixed by the Frobenius q automorphism (aij ) 7→ (aij ). 4 It is easy to prove that G is a finite group. Chevalley groups

The Finite Simple Groups In 1955 Claude Chevalley found a way to construct (some of) I: Description the finite classical groups using the Lie theory, rather than the Nick Gill (OU) theory of forms. 1 Let G be a simple algebraic subgroup of GLn(Fq). 2 The possibilities for G are given by the Dynkin diagrams. F 3 Look at G = G , the subset of G fixed by the Frobenius q automorphism (aij ) 7→ (aij ). 4 It is easy to prove that G is a finite group. These groups are called the Chevalley groups. They include most of the finite classical groups, plus some more - the first exceptional groups:

G2(q), F4(q), E6(q), E7(q), E8(q). In 1959 Robert Steinberg realised that the ‘missing’ classical groups could be constructed by twisting Chevalley groups using a graph automorphism.

Exceptional and twisted I: Steinberg groups

The Finite What about PSU(q)? Simple Groups I: Description

Nick Gill (OU) 1 Let G be a simple algebraic subgroup of GLn(Fq). 2 Suppose that the Dynkin diagram for G has no multiple bonds and admits a symmetry. 3 We can modify the Frobenius automorphism using this symmetry to construct a different finite subgroup of G. These are the Steinberg groups. They include the missing finite classical groups, plus some more exceptional groups: 3 2 D4(q), E6(q).

Exceptional and twisted I: Steinberg groups

The Finite What about PSU(q)? Simple Groups I: Description

Nick Gill (OU) In 1959 Robert Steinberg realised that the ‘missing’ classical groups could be constructed by twisting Chevalley groups using a graph automorphism. 2 Suppose that the Dynkin diagram for G has no multiple bonds and admits a symmetry. 3 We can modify the Frobenius automorphism using this symmetry to construct a different finite subgroup of G. These are the Steinberg groups. They include the missing finite classical groups, plus some more exceptional groups: 3 2 D4(q), E6(q).

Exceptional and twisted I: Steinberg groups

The Finite What about PSU(q)? Simple Groups I: Description

Nick Gill (OU) In 1959 Robert Steinberg realised that the ‘missing’ classical groups could be constructed by twisting Chevalley groups using a graph automorphism.

1 Let G be a simple algebraic subgroup of GLn(Fq). 3 We can modify the Frobenius automorphism using this symmetry to construct a different finite subgroup of G. These are the Steinberg groups. They include the missing finite classical groups, plus some more exceptional groups: 3 2 D4(q), E6(q).

Exceptional and twisted I: Steinberg groups

The Finite What about PSU(q)? Simple Groups I: Description

Nick Gill (OU) In 1959 Robert Steinberg realised that the ‘missing’ classical groups could be constructed by twisting Chevalley groups using a graph automorphism.

1 Let G be a simple algebraic subgroup of GLn(Fq). 2 Suppose that the Dynkin diagram for G has no multiple bonds and admits a symmetry. These are the Steinberg groups. They include the missing ﬁnite classical groups, plus some more exceptional groups: 3 2 D4(q), E6(q).

Exceptional and twisted I: Steinberg groups

The Finite What about PSU(q)? Simple Groups I: Description

Nick Gill (OU) In 1959 Robert Steinberg realised that the ‘missing’ classical groups could be constructed by twisting Chevalley groups using a graph automorphism.

The Finite What about PSU(q)? Simple Groups I: Description

Nick Gill (OU) In 1959 Robert Steinberg realised that the ‘missing’ classical groups could be constructed by twisting Chevalley groups using a graph automorphism.

1 Let G be a simple algebraic subgroup of GLn(Fq). 2 Suppose that the Dynkin diagram for G has no multiple bonds and admits a symmetry. 3 We can modify the Frobenius automorphism using this symmetry to construct a different finite subgroup of G. These are the Steinberg groups. They include the missing finite classical groups, plus some more exceptional groups: 3 2 D4(q), E6(q). In 1960, using totally different ideas, Michio Suzuki constructed a new infinite family of simple groups – Sz(22a+1) – as 4 × 4 matrices over a field of even order. Soon after, Rimhak Ree realised that Steinberg’s ‘twist’ could be applied to algebraic groups G whose Dynkin diagram was symmetric and included multiple bonds. He obtained the Ree groups. They include the Suzuki groups, as well as two new families of exceptional groups:

2 2a+1 2a+1 2 2a+1 2 2a+1 B2(2 ) = Sz(2 ), F4(2 ), G2(3 ).

If a > 1 these groups are all simple. In 1964 Tits showed that 2 F4(2) contains a new simple group as a subgroup of index 2. This is the last simple group of Lie type.

Exceptional and twisted II: Ree-Suzuki

The Finite Simple Groups I: Description

Nick Gill (OU) Soon after, Rimhak Ree realised that Steinberg’s ‘twist’ could be applied to algebraic groups G whose Dynkin diagram was symmetric and included multiple bonds. He obtained the Ree groups. They include the Suzuki groups, as well as two new families of exceptional groups:

2 2a+1 2a+1 2 2a+1 2 2a+1 B2(2 ) = Sz(2 ), F4(2 ), G2(3 ).

If a > 1 these groups are all simple. In 1964 Tits showed that 2 F4(2) contains a new simple group as a subgroup of index 2. This is the last simple group of Lie type.

Exceptional and twisted II: Ree-Suzuki

The Finite Simple Groups In 1960, using totally different ideas, Michio I: Description Suzuki constructed a new infinite family of Nick Gill 2a+1 (OU) simple groups – Sz(2 ) – as 4 × 4 matrices over a field of even order. He obtained the Ree groups. They include the Suzuki groups, as well as two new families of exceptional groups:

2 2a+1 2a+1 2 2a+1 2 2a+1 B2(2 ) = Sz(2 ), F4(2 ), G2(3 ).

If a > 1 these groups are all simple. In 1964 Tits showed that 2 F4(2) contains a new simple group as a subgroup of index 2. This is the last simple group of Lie type.

Exceptional and twisted II: Ree-Suzuki

The Finite Simple Groups In 1960, using totally different ideas, Michio I: Description Suzuki constructed a new infinite family of Nick Gill 2a+1 (OU) simple groups – Sz(2 ) – as 4 × 4 matrices over a field of even order. Soon after, Rimhak Ree realised that Steinberg’s ‘twist’ could be applied to algebraic groups G whose Dynkin diagram was symmetric and included multiple bonds. If a > 1 these groups are all simple. In 1964 Tits showed that 2 F4(2) contains a new simple group as a subgroup of index 2. This is the last simple group of Lie type.

Exceptional and twisted II: Ree-Suzuki

The Finite Simple Groups In 1960, using totally different ideas, Michio I: Description Suzuki constructed a new infinite family of Nick Gill 2a+1 (OU) simple groups – Sz(2 ) – as 4 × 4 matrices over a field of even order. Soon after, Rimhak Ree realised that Steinberg’s ‘twist’ could be applied to algebraic groups G whose Dynkin diagram was symmetric and included multiple bonds. He obtained the Ree groups. They include the Suzuki groups, as well as two new families of exceptional groups:

2 2a+1 2a+1 2 2a+1 2 2a+1 B2(2 ) = Sz(2 ), F4(2 ), G2(3 ). Exceptional and twisted II: Ree-Suzuki

The Finite Simple Groups In 1960, using totally different ideas, Michio I: Description Suzuki constructed a new infinite family of Nick Gill 2a+1 (OU) simple groups – Sz(2 ) – as 4 × 4 matrices over a field of even order. Soon after, Rimhak Ree realised that Steinberg’s ‘twist’ could be applied to algebraic groups G whose Dynkin diagram was symmetric and included multiple bonds. He obtained the Ree groups. They include the Suzuki groups, as well as two new families of exceptional groups:

2 2a+1 2a+1 2 2a+1 2 2a+1 B2(2 ) = Sz(2 ), F4(2 ), G2(3 ).

If a > 1 these groups are all simple. In 1964 Tits showed that 2 F4(2) contains a new simple group as a subgroup of index 2. This is the last simple group of Lie type. One year later, in 1965, Zvonimir Janko constructed the ﬁrst new sporadic simple group – J1 – since Mathieu in 1861.

But that is another story...

The rise of the sporadics

The Finite Simple Groups I: Description

Nick Gill (OU) But that is another story...

The rise of the sporadics

The Finite Simple Groups I: Description

Nick Gill (OU)

One year later, in 1965, Zvonimir Janko constructed the ﬁrst new sporadic simple group – J1 – since Mathieu in 1861. The rise of the sporadics

The Finite Simple Groups I: Description

Nick Gill (OU)

One year later, in 1965, Zvonimir Janko constructed the ﬁrst new sporadic simple group – J1 – since Mathieu in 1861.

But that is another story...