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 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; 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: 1 the cyclic groups of prime order; 2 the alternating groups of degree at least 5; 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: 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 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 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 8α; β 2 Ω; 9g 2 G; αg = β: We say that G is k-transitive if ∗k 8(α1; : : : ; αk ); (β1; : : : ; βk ) 2 Ω ; 9g 2 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 8α; β 2 Ω; 9g 2 G; αg = β: We say that G is k-transitive if ∗k 8(α1; : : : ; αk ); (β1; : : : ; βk ) 2 Ω ; 9g 2 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 8(α1; : : : ; αk ); (β1; : : : ; βk ) 2 Ω ; 9g 2 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 8α; β 2 Ω; 9g 2 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 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 8α; β 2 Ω; 9g 2 G; αg = β: We say that G is k-transitive if ∗k 8(α1; : : : ; αk ); (β1; : : : ; βk ) 2 Ω ; 9g 2 G; (α1g; α2g; : : : ; αk g) = (β1; : : : ; βk ): 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 8α; β 2 Ω; 9g 2 G; αg = β: We say that G is k-transitive if ∗k 8(α1; : : : ; αk ); (β1; : : : ; βk ) 2 Ω ; 9g 2 G; (α1g; α2g; : : : ; αk g) = (β1; : : : ; βk ): k-transitive groups have a tendency towards simplicity..