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Automorphism group of a group

1,304 bytes added, 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 <math>S_n</math> is a [[complete group]] if <math>n \ne 2,6</math>.
|}
 
===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 <math>n</math>, it is an abelian group of order <math>\varphi(n)</math> defined as the [[multiplicative group modulo n]]. It is itself cyclic if <math>n = 2,4</math>, 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]]
|}
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