Automorphism of $S_4$

Question

How can we show that every automorphism of $S_4$ is an inner automorphism ?

Answer 1

Note that:

• Any automorphism preserves the conjugacy class of transpositions.( By counting size of conjugacy classes)

• Any automorphism preserves whether two transpositions share an element (by looking at the order of their product)

• Any automorphism permutes the four classes of transpositions {those that move 1}, {those that move 2}, {those that move 3}, {those that move 4}

• This gives you a permutation on {1,2,3,4} and it is not hard to show that your automorphism is conjugation by this permutation

Answer 2

Automorphisms send conjugacy classes to conjugacy classes and preserve elements' order. In $S_4$, distinct conjugacy classes have elements of distinct orders except the conjugacy classes of the transpositions and the double transpositions (both whose elements have order $2$); but the former has size $6$, whilst the latter has size $3$, so no one automorphism can swap them. Therefore, every automorphism is cycle type-preserving, and then inner.