Symmetric Group - Relation With Alternating Group

Relation With Alternating Group

For n≥5, the alternating group An is simple, and the induced quotient is the sign map: AnSnS2 which is split by taking a transposition of two elements. Thus Sn is the semidirect product AnS2, and has no other proper normal subgroups, as they would intersect An in either the identity (and thus themselves be the identity or a 2-element group, which is not normal), or in An (and thus themselves be An or Sn).

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

Conversely, for n ≠ 6, Sn 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, Sn is an almost simple group, as it lies between the simple group An and its group of automorphisms.

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