By Peter J. Cameron

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Extra resources for Notes on finite group theory [Lecture notes]

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An automorphism which is not inner is called outer. ] — but is defined to be the quotient group Aut(G)/ Inn(G). Thus, the outer automorphism group of G is trivial if and only if G has no outer automorphisms. 54 CHAPTER 2. 6 Outer automorphisms of S6 For n ≥ 3, Z(Sn ) = {1}, so Inn(Sn ) ∼ = Sn . It turns out that, except for the single value n = 6, we actually have Aut(Sn ) = Sn , so that Sn has no outer automorphisms. We will not prove that, but will construct outer automorphisms of S6 in two different ways, and hence show that | Out(S6 )| = 2.

An equivalence class of a congruence is called a block. Note that, if B is a block, then so is Bg for any g ∈ G. There are always two trivial congruences: equality: α ≡ β if and only if α = β ; the universal relation: α ≡ β for all α, β ∈ Ω. The action is called imprimitive if there is a non-trivial congruence, and primitive if not. Example Let G be the symmetry group of a square (the dihedral group of order 8), acting on Ω, the set of four vertices of the square. The relation ≡ defined by α ≡ β if α and β are equal or opposite is a congruence, with two blocks of size 2.

3 Let G be the symmetry group of the cube. Show that the action of G on the set of vertices of the cube is transitive but imprimitive, and describe all the congruences. Repeat for the action of G on the set of faces, and on the set of edges. 4 An automorphism of a group G is an isomorphism from G to itself. An inner automorphism of G is a conjugation map, one of the form cg : x → g−1 xg. (a) Show that the set of automorphisms, with the operation of conjugation, is a group Aut(G). (b) Show that the set of inner automorphisms is a subgroup Inn(G) of Aut(G).

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