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Automorphism group of a group

, 03:48, 28 May 2013
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==Particular cases==

===Particular groups===

{| class="sortable" border="1"
|-
| [[symmetric group:S4]] || 24 || [[symmetric group:S4]] || 24 || [[endomorphism structure of symmetric group:S4]] || [[symmetric groups are complete]]: the symmetric group $S_n$ is a [[complete group]] if $n \ne 2,6$.
|}

===Group families===

For various group families, the automorphism group can be described in terms of parameters for members of the families. The descriptions are sometimes quite complicated, so we simply provide links:

{| class="sortable" border="1"
! Family !! Description of automorphism group !! Endomorphism structure information
|-
| [[finite cyclic group]] || For a cyclic group of order $n$, it is an abelian group of order $\varphi(n)$ defined as the [[multiplicative group modulo n]]. It is itself cyclic if $n = 2,4$, a power of an odd prime, or twice a power of an odd primes || [[endomorphism structure of finite cyclic groups]]
|-
| [[finite abelian group]] || (no simple description) || --
|-
| [[symmetric group]] || the same [[symmetric group]] if the degree is not 2 or 6. For degree 2, the [[trivial group]]. For degree 6 (i.e., [[symmetric group:S6]]), the group [[automorphism group of alternating group:A6]]. || [[endomorphism structure of symmetric groups]]
|-
| [[alternating group]] || the [[symmetric group]] if the degree is at least 3 and not equal to 6. For degree 6 (i.e., [[alternating group:A6]]), the group [[automorphism group of alternating group:A6]]. || [[endomorphism structure of alternating groups]]
|}