Editing Alternating group (section)

== Automorphism group ==
{{details|Automorphisms of the symmetric and alternating groups}}
{| align="right" class=wikitable
|-
! ''n''
! Aut(A<sub>''n''</sub>)
! Out(A<sub>''n''</sub>)
|- align=center
|''n'' ≥ 4, ''n'' ≠ 6
| S<sub>''n''</sub>
| Z<sub>2</sub>
|- align=center
| ''n'' = 1, 2
| Z<sub>1</sub>
| Z<sub>1</sub>
|- align=center
| ''n'' = 3
| Z<sub>2</sub>
| Z<sub>2</sub>
|- align=center
| ''n'' = 6
| S<sub>6</sub> ⋊ Z<sub>2</sub>
| V = Z<sub>2</sub> × Z<sub>2</sub>
|}

For {{nowrap|''n'' > 3}}, except for {{nowrap|1=''n'' = 6}}, the  [[automorphism group]] of A<sub>''n''</sub> is the symmetric group S<sub>''n''</sub>, with [[inner automorphism group]] A<sub>''n''</sub> and [[outer automorphism group]] Z<sub>2</sub>; the outer automorphism comes from conjugation by an odd permutation.

For {{nowrap|1=''n'' = 1}} and 2, the automorphism group is trivial. For {{nowrap|1=''n'' = 3}} the automorphism group is Z<sub>2</sub>, with trivial inner automorphism group and outer automorphism group Z<sub>2</sub>.

The outer automorphism group of A<sub>6</sub> is [[Klein four-group|the Klein four-group]] {{nowrap|1=V = Z<sub>2</sub> × Z<sub>2</sub>}}, and is related to [[Symmetric group#Automorphism group|the outer automorphism of S<sub>6</sub>]]. The extra outer automorphism in A<sub>6</sub> swaps the 3-cycles (like&nbsp;(123)) with elements of shape 3<sup>2</sup> (like&nbsp;{{nowrap|(123)(456)}}).