Editing Outer automorphism group

{{Short description|Mathematical group}}
In [[mathematics]], the '''outer automorphism group''' of a [[group (mathematics)|group]], {{mvar|G}}, is the [[quotient group|quotient]], {{math|Aut(''G'')&nbsp;/&nbsp;Inn(''G'')}}, where {{math|Aut(''G'')}} is the [[automorphism group]] of {{mvar|G}} and {{math|Inn(''G''}}) is the subgroup consisting of [[inner automorphism]]s. The outer automorphism group is usually denoted {{math|Out(''G'')}}. If {{math|Out(''G'')}} is trivial and {{mvar|G}} has a trivial [[Center (group theory)|center]], then {{mvar|G}} is said to be [[complete group|complete]].

An automorphism of a group that is not inner is called an '''outer automorphism'''.<ref>Despite the name, these do not form the elements of the outer automorphism group.  For this reason, the term ''non-inner automorphism'' is sometimes preferred.</ref> The [[coset|cosets]] of {{math|Inn(''G'')}} with respect to outer automorphisms are then the elements of {{math|Out(''G'')}}; this is an instance of the fact that quotients of groups are not, in general, (isomorphic to) subgroups. If the inner automorphism group is trivial (when a group is abelian), the automorphism group and outer automorphism group are naturally identified; that is, the outer automorphism group does act on the group.

For example, for the [[alternating group]], {{math|A{{sub|''n''}}}}, the outer automorphism group is usually the group of order 2, with exceptions noted below. Considering {{math|A{{sub|''n''}}}} as a subgroup of the [[symmetric group]], {{math|S{{sub|''n''}}}}, conjugation by any [[odd permutation]] is an outer automorphism of {{math|A{{sub|''n''}}}} or more precisely "represents the class of the (non-trivial) outer automorphism of {{math|A{{sub|''n''}}}}", but the outer automorphism does not correspond to conjugation by any ''particular'' odd element, and all conjugations by odd elements are equivalent up to conjugation by an even element.

== Structure ==
The [[Schreier conjecture]] asserts that {{math|Out(''G'')}} is always a [[solvable group]] when {{mvar|G}} is a finite [[simple group]]. This result is now known to be true as a corollary of the [[classification of finite simple groups]], although no simpler proof is known.

== As dual of the center ==
The outer automorphism group is [[duality (mathematics)|dual]] to the center in the following sense: conjugation by an element of {{mvar|G}} is an automorphism, yielding a map {{math|''σ'' : ''G'' → Aut(''G'')}}. The [[kernel (algebra)|kernel]] of the conjugation map is the center, while the [[cokernel]] is the outer automorphism group (and the image is the [[inner automorphism]] group). This can be summarized by the [[exact sequence]]

<math display="block">Z(G) \hookrightarrow G \, \overset{\sigma}{\longrightarrow} \, \mathrm{Aut}(G) \twoheadrightarrow \mathrm{Out}(G)</math>

== Applications ==
The outer automorphism group of a group acts on [[conjugacy class]]es, and accordingly on the [[character table]]. See details at [[Character table#Outer automorphisms|character table: outer automorphisms]].

=== Topology of surfaces ===
The outer automorphism group is important in the [[topology]] of [[surface (topology)|surface]]s because there is a connection provided by the [[Dehn&ndash;Nielsen theorem]]: the extended [[mapping class group]] of the surface is the outer automorphism group of its [[fundamental group]].

== 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}}

== In symmetric and alternating groups ==
{{details|Automorphisms of the symmetric and alternating groups}}

The outer automorphism group of a finite simple group in some infinite family of finite simple groups can almost always be given by a uniform formula that works for all elements of the family. There is just one exception to this:<ref>ATLAS p. xvi</ref> the alternating group {{math|A{{sub|6}}}} has outer automorphism group of order 4, rather than 2 as do the other simple alternating groups (given by conjugation by an [[odd permutation]]). Equivalently the symmetric group {{math|S{{sub|6}}}} is the only symmetric group with a non-trivial outer automorphism group.
: <math>\begin{align}
             n \neq 6: \operatorname{Out}(\mathrm{S}_n) & = \mathrm{C}_1 \\
  n \geq 3,\ n \neq 6: \operatorname{Out}(\mathrm{A}_n) & = \mathrm{C}_2 \\
                       \operatorname{Out}(\mathrm{S}_6) & = \mathrm{C}_2 \\
                       \operatorname{Out}(\mathrm{A}_6) & = \mathrm{C}_2 \times \mathrm{C}_2
\end{align}</math>

Note that, in the case of {{math|''G'' {{=}} A{{sub|6}} {{=}} PSL(2, 9)}}, the sequence {{math|1 ⟶ ''G'' ⟶ Aut(''G'') ⟶ Out(''G'') ⟶ 1}} does not split. A similar result holds for any {{math|PSL(2, ''q''{{sup|2}})}}, {{mvar|q}} odd.

== In reductive algebraic groups ==
[[File:Dynkin diagram D4.png|thumb|150px|The symmetries of the [[Dynkin diagram]], {{math|D{{sub|4}}}}, correspond to the outer automorphisms of {{math|Spin(8)}} in triality.]]
Let {{mvar|G}} now be a connected [[reductive group]] over an [[algebraically closed field]]. Then any two [[Borel subgroup]]s are conjugate by an inner automorphism, so to study outer automorphisms it suffices to consider automorphisms that fix a given Borel subgroup. Associated to the Borel subgroup is a set of [[Root system#Positive roots and simple roots|simple roots]], and the outer automorphism may permute them, while preserving the structure of the associated [[Root system#Classification of root systems by Dynkin diagrams|Dynkin diagram]]. In this way one may identify the automorphism group of the Dynkin diagram of {{mvar|G}} with a subgroup of {{math|Out(''G'')}}.

{{math|D{{sub|4}}}} has a very symmetric Dynkin diagram, which yields a large outer automorphism group of {{math|[[Spin(8)]]}}, namely {{math|Out(Spin(8)) {{=}} S{{sub|3}}}}; this is called [[triality]].

== In complex and real simple Lie algebras ==
The preceding interpretation of outer automorphisms as symmetries of a Dynkin diagram follows from the general fact, that for a complex or real simple Lie algebra, {{mvar|𝔤}}, the automorphism group {{math|Aut(''𝔤'')}} is a [[semidirect product]] of {{math|Inn(''𝔤'')}} and {{math|Out(''𝔤'')}}; i.e., the [[exact sequence|short exact sequence]]
: {{math|1 ⟶ Inn(''𝔤'') ⟶ Aut(''𝔤'') ⟶ Out(''𝔤'') ⟶ 1}}
splits. In the complex simple case, this is a classical result,<ref>{{citation| last1         = Fulton
| first1        = William
| author1-link  = William Fulton (mathematician)
| last2        = Harris
| first2       = Joe
| author2-link = Joe Harris (mathematician)
| year         = 1991
| title        = Representation theory. A first course
| publisher    = Springer-Verlag
| location     = New York
| series       = [[Graduate Texts in Mathematics]], Readings in Mathematics
| volume       = 129
| isbn         = 978-0-387-97495-8
| doi          = 10.1007/978-1-4612-0979-9
| oclc         = 246650103
| language     = en-gb
| mr           = 1153249|contribution=Proposition D.40}}</ref> whereas for real simple Lie algebras, this fact was proven as recently as 2010.<ref name="JOLT">[http://www.heldermann.de/JLT/JLT20/JLT204/jlt20035.htm JLT20035]</ref>

== Word play ==
The term ''outer automorphism'' lends itself to [[word play]]: the term ''outermorphism'' is sometimes used for ''outer automorphism'', and a particular [[geometric group action|geometry]] on which {{math|Out(''F''{{sub|''n''}})}} acts is called ''[[Out(Fn)#Outer space|outer space]]''.

== See also ==
* [[Mapping class group]]
* [[Out(Fn)|Out(''F{{sub|n}}'')]]
* [[Outer space (mathematics)|Outer space]]

== References ==
{{More citations needed|date=November 2009}}
{{reflist}}

== External links ==
* [http://brauer.maths.qmul.ac.uk/Atlas/v3/ ATLAS of Finite Group Representations-V3], contains a lot of information on various classes of finite groups (in particular sporadic simple groups), including the order of {{math|Out(''G'')}} for each group listed.

[[Category:Group theory]]
[[Category:Group automorphisms]]