Editing Symmetric group (section)

== Relation with alternating group ==
For {{nowrap|''n'' ≥ 5}}, the [[alternating group]] A<sub>''n''</sub> is [[Simple group|simple]], and the induced quotient is the sign map: {{nowrap|A<sub>''n''</sub> → S<sub>''n''</sub> → S<sub>2</sub>}} which is split by taking a transposition of two elements. Thus S<sub>''n''</sub> is the semidirect product {{nowrap|A<sub>''n''</sub> ⋊ S<sub>2</sub>}}, and has no other proper normal subgroups, as they would intersect A<sub>''n''</sub> in either the identity (and thus themselves be the identity or a 2-element group, which is not normal), or in A<sub>''n''</sub> (and thus themselves be A<sub>''n''</sub> or S<sub>''n''</sub>).

S<sub>''n''</sub> acts on its subgroup A<sub>''n''</sub> by conjugation, and for {{nowrap|''n'' ≠ 6}}, S<sub>''n''</sub> is the full automorphism group of A<sub>''n''</sub>: Aut(A<sub>''n''</sub>) ≅ S<sub>''n''</sub>. Conjugation by even elements are [[inner automorphism]]s of A<sub>''n''</sub> while the [[outer automorphism]] of A<sub>''n''</sub> of order 2 corresponds to conjugation by an odd element. For {{nowrap|1=''n'' = 6}}, there is an [[Automorphisms of the symmetric and alternating groups#exceptional outer automorphism|exceptional outer automorphism]] of A<sub>''n''</sub> so S<sub>''n''</sub> is not the full automorphism group of A<sub>''n''</sub>.

Conversely, for {{nowrap|''n'' ≠ 6}}, S<sub>''n''</sub> has no outer automorphisms, and for {{nowrap|''n'' ≠ 2}} it has no center, so for {{nowrap|''n'' ≠ 2, 6}} it is a [[complete group]], as discussed in [[#Automorphism group|automorphism group]], below.

For {{nowrap|''n'' ≥ 5}}, S<sub>''n''</sub> is an [[almost simple group]], as it lies between the simple group A<sub>''n''</sub> and its group of automorphisms.

S<sub>''n''</sub> can be embedded into A<sub>''n''+2</sub> by appending the transposition {{nowrap|(''n'' + 1, ''n'' + 2)}} to all odd permutations, while embedding into A<sub>''n''+1</sub> is impossible for {{nowrap|''n'' > 1}}.