S n acts on its subgroup A n by conjugation, and for n ≠ 6, S n is the full automorphism group of A n : Aut(A n) ≅ S n. Conjugation by even elements are inner automorphisms of A n while the outer automorphism of A n of order 2 corresponds to conjugation by an odd element. For n = 6, there is an exceptional outer automorphism of A n so S n is not the full automorphism group of A n.