Automorphism of $S_4$

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

Solutions Collecting From Web of "Automorphism of $S_4$"

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