Editing Outer automorphism group (section)

== In finite groups ==
For the outer automorphism groups of all finite simple groups see the [[list of finite simple groups]]. Sporadic simple groups and alternating groups (other than the alternating group, {{math|A{{sub|6}}}}; see below) all have outer automorphism groups of order 1 or 2. The outer automorphism group of a finite simple [[group of Lie type]] is an extension of a group of "diagonal automorphisms" (cyclic except for {{math|[[list of finite simple groups#Dn.28q.29 n .3E 3 Chevalley groups.2C orthogonal groups|D{{sub|''n''}}(''q'')]]}}, when it has order 4), a group of "field automorphisms" (always cyclic), and a group of "graph automorphisms" (of order 1 or 2 except for {{math|D{{sub|4}}(''q'')}}, when it is the symmetric group on 3 points). These extensions are not always [[semidirect product]]s, as the case of the alternating group {{math|A{{sub|6}}}} shows; a precise criterion for this to happen was given in 2003.<ref>A. Lucchini, F. Menegazzo, M. Morigi (2003), "[https://projecteuclid.org/download/pdf_1/euclid.ijm/1258488162 On the existence of a complement for a finite simple group in its automorphism group]", ''Illinois J. Math.'' 47, 395–418.</ref>

{| class="wikitable"
|-
! Group
! Parameter
! {{math|Out(''G'')}}
! {{math|{{abs|Out(''G'')}}}}
|-
| {{math|[[Infinite cyclic group|Z]]}}
| || {{math|[[cyclic group|C{{sub|2}}]]}}
| {{math|2}}: the identity and the outer automorphism {{math|''x'' ↦ −''x''}}
|-
| {{math|[[cyclic group|C{{sub|''n''}}]]}} || {{math|''n'' > 2}} 
| {{math|[[Multiplicative group of integers modulo n|(ℤ/''n''ℤ){{sup|×}}]]}}
| {{math|[[Euler's totient function|''φ''(''n'')]] {{=}} }}<math>n\prod_{p|n}\left(1 - \frac{1}{p}\right)</math>; one corresponding to multiplication by an invertible element in the [[Ring (mathematics)|ring]] {{math|ℤ/''n''ℤ}}.
|-
| {{math|[[cyclic group|Z{{sub|''p''}}{{sup|''n''}}]]}}
| {{mvar|p}} prime, {{math|''n'' > 1}}
| {{math|[[general linear group|GL{{sub|''n''}}(''p'')]]}}
| {{math|(''p''{{sup|''n''}} − 1)(''p''{{sup|''n''}} − ''p'' )(''p''{{sup|''n''}} − ''p''{{sup|2}})...(''p''{{sup|''n''}} − ''p''{{sup|''n''−1}})}}
|-
| {{math|[[symmetric group|S{{sub|''n''}}]]}}
| {{math|''n'' ≠ 6}} || {{math|[[Trivial group|C{{sub|1}}]]}}
| {{math|1}}
|-
| {{math|[[symmetric group|S{{sub|6}}]]}}
| &nbsp; || {{math|C{{sub|2}}}} (see below)
| {{math|2}}
|-
| {{math|[[alternating group|A{{sub|''n''}}]]}}
| {{math|''n'' ≠ 6}} || {{math|C{{sub|2}}}}
| {{math|2}}
|-
| {{math|[[alternating group|A{{sub|6}}]]}}
| &nbsp;
| {{math|[[Klein four-group|C{{sub|2}} × C{{sub|2}}]]}} (see below)
| {{math|4}}
|-
| {{math|[[projective special linear group|PSL{{sub|2}}(''p'')]]}}
| {{math|''p'' > 3}} prime || {{math|C{{sub|2}}}}
| {{math|2}}
|-
| {{math|[[projective special linear group|PSL{{sub|2}}(2{{sup|''n''}})]]}}
| {{math|''n'' > 1}} || {{math|C{{sub|''n''}}}}
| {{mvar|n}}
|-
| {{math|[[projective special linear group|PSL{{sub|3}}(4)]] {{=}} [[Mathieu group|M{{sub|21}}]]}}
| &nbsp; || {{math|[[dihedral group of order 6|Dih{{sub|6}}]]}}
| {{math|12}}
|-
| {{math|[[Mathieu group|M{{sub|''n''}}]]}}
| {{math|''n'' ∈ {11, 23, 24} }} || {{math|C{{sub|1}}}}
| {{math|1}}
|-
| {{math|[[Mathieu group|M{{sub|''n''}}]]}}
| {{math|''n'' ∈ {12, 22} }} || {{math|C{{sub|2}}}}
| {{math|2}}
|-
| {{math|[[Conway group|Co{{sub|''n''}}]]}}
| {{math|''n'' ∈ {1, 2, 3} }} || {{math|C{{sub|1}}}}
| {{math|1}}
|}{{Citation needed|date=February 2007}}