Symmetric Group - Relation With Alternating Group

Relation With Alternating Group

For n≥5, the alternating group A_n is simple, and the induced quotient is the sign map: A_n → S_n → S₂ which is split by taking a transposition of two elements. Thus S_n is the semidirect product A_n ⋊ S₂, and has no other proper normal subgroups, as they would intersect A_n in either the identity (and thus themselves be the identity or a 2-element group, which is not normal), or in A_n (and thus themselves be A_n or S_n).

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.

Conversely, for n ≠ 6, S_n has no outer automorphisms, and for n ≠ 2 it has no center, so for n ≠ 2, 6 it is a complete group, as discussed in automorphism group, below.

For n ≥ 5, S_n is an almost simple group, as it lies between the simple group A_n and its group of automorphisms.