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| In [[mathematics]], the '''outer automorphism group''' of a [[group (mathematics)|group]] ''G''
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| is the [[quotient group|quotient]] Aut(''G'') / Inn(''G''), where Aut(''G'') is the [[automorphism group]] of ''G'' and Inn(''G'') is the subgroup consisting of [[inner automorphism]]s. The outer automorphism group is usually denoted Out(''G''). If Out(''G'') is trivial and ''G'' has a trivial center, then ''G'' is said to be [[complete group|complete]].
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| An automorphism of a group which is not inner is called an outer automorphism. Note that the elements of Out(''G'') are cosets of automorphisms of ''G'', and not themselves automorphisms; this is an instance of the fact that quotients of groups are not in general (isomorphic to) subgroups. Elements of Out(''G'') are cosets of Inn(''G'') in Aut(''G'').
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| For example, for the [[alternating group]] ''A''<sub>''n''</sub>, the outer automorphism group is usually the group of order 2, with exceptions noted below. Considering ''A''<sub>''n''</sub> as a subgroup of the [[symmetric group]] ''S''<sub>''n''</sub> conjugation by any [[odd permutation]] is an outer automorphism of ''A''<sub>''n''</sub> or more precisely "represents the class of the (non-trivial) outer automorphism of ''A''<sub>''n''</sub>", 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.
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| However, for an abelian group ''A,'' the inner automorphism group is trivial and thus the automorphism group and outer automorphism group are naturally identified, and outer automorphisms do act on ''A''.
| | '''MathML''' |
| | :<math forcemathmode="mathml">E=mc^2</math> |
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| ==Out(''G'') for some finite groups==
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| | :<math forcemathmode="png">E=mc^2</math> |
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| 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 ''A''<sub>6</sub>; 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 [[List of finite simple groups#Dn.28q.29 n .3E 3 Chevalley groups.2C orthogonal groups|D<sub>''n''</sub>(''q'')]] when it has order 4), a group of "field automorphisms" (always cyclic), and
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| a group of "graph automorphisms" (of order 1 or 2 except for D<sub>4</sub>(''q'') when it is the symmetric group on 3 points). These extensions are [[semidirect product]]s except that for the [[group of Lie type#Suzuki–Ree groups|Suzuki-Ree groups]] the graph automorphism squares to a generator of the field automorphisms.
| | :<math forcemathmode="source">E=mc^2</math> --> |
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| {| class="wikitable"
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| |-
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| ! Group
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| ! Parameter
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| ! Out(G)
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| ! <math>|\mbox{Out}(G)|</math>
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| |-
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| | [[Integer|''Z'']]
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| | infinite cyclic || Z<sub>2</sub>
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| | 2; the identity and the map f(x) = -x
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| |-
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| | [[cyclic group|Z<sub>''n''</sub>]] || ''n'' > 2
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| | [[Multiplicative group of integers modulo n|Z<sub>''n''</sub><sup>×]]</sup>
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| |[[Euler's totient function|φ(n)]] = <math>n\prod_{p|n}\left(1-\frac{1}{p}\right)</math> elements; one corresponding to multiplication by an invertible element in Z<sub>''n''</sub> viewed as a ring.
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| |-
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| | [[cyclic group|Z<sub>''p''</sub><sup>''n''</sup>]]
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| | ''p'' prime, ''n'' > 1
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| | [[general linear group|GL<sub>''n''</sub>(''p'')]]
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| |(''p''<sup>''n''</sup> − 1)(''p''<sup>''n''</sup> − ''p'' )(''p''<sup>''n''</sup> − ''p''<sup>2</sup>) ... (''p''<sup>''n''</sup> − ''p''<sup>''n''−1</sup>)
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| elements
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| |-
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| | [[symmetric group|''S''<sub>''n''</sub>]]
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| | n ≠ 6 || [[trivial group|trivial]]
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| | 1
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| |-
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| | [[symmetric group|''S''<sub>6</sub>]]
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| | || Z<sub>2</sub> (see below)
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| | 2
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| |-
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| | [[alternating group|''A''<sub>''n''</sub>]]
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| | ''n'' ≠ 6 || Z<sub>2</sub>
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| | 2
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| |-
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| | [[alternating group|''A''<sub>6</sub>]]
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| | Z<sub>2</sub> × Z<sub>2</sub>(see below)
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| | 4
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| |-
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| | [[projective special linear group|PSL<sub>2</sub>(''p'')]]
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| | ''p'' > 3 prime || Z<sub>2</sub>
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| |2
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| |-
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| | [[projective special linear group|PSL<sub>2</sub>(2<sup>''n''</sup>)]]
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| | ''n'' > 1 || Z<sub>''n''</sub>
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| |''n''
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| |-
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| | [[projective special linear group|PSL<sub>3</sub>(4)]] = [[Mathieu group|M<sub>21</sub>]]
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| | || [[dihedral group|Dih<sub>6</sub>]]
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| | 12
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| |-
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| | [[Mathieu group|M<sub>''n''</sub>]]
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| | ''n'' = 11, 23, 24 || [[trivial group|trivial]]
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| |1
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| |-
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| | [[Mathieu group|M<sub>''n''</sub>]]
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| | ''n'' = 12, 22 || Z<sub>2</sub>
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| |2
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| |-
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| | [[Conway group|Co<sub>''n''</sub>]]
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| | ''n'' = 1, 2, 3 || [[trivial group|trivial]]
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| |1
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| |}{{Citation needed|date=February 2007}}
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| == The outer automorphisms of the symmetric and alternating groups== | | ==Demos== |
| {{details|Automorphisms of the symmetric and alternating groups}}
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| 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 ''A''<sub>6</sub> 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 ''S''<sub>6</sub> is the only symmetric group with a non-trivial outer automorphism group.
| | Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]: |
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| :<math>
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| \begin{align}
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| n\neq 6: \mathrm{Out}(S_n) & = 1 \\
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| n\geq 3,\ n\neq 6: \mathrm{Out}(A_n) & = C_2 \\
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| \mathrm{Out}(S_6) & = C_2 \\
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| \mathrm{Out}(A_6) & = C_2 \times C_2
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| \end{align}
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| </math>
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| ==Outer automorphism groups of complex Lie groups==
| | * accessibility: |
| [[File:Dynkin diagram D4.png|thumb|150px|The symmetries of the [[Dynkin diagram]] D<sub>4</sub> correspond to the outer automorphisms of Spin(8) in triality.]] | | ** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]] |
| Let ''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 ''G'' with a subgroup of Out(''G'').
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| | ** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]]. |
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| ''D''<sub>4</sub> has a very symmetric Dynkin diagram, which yields a large outer automorphism group of [[Spin(8)]], namely Out(Spin(8)) = ''S''<sub>3</sub>; this is called [[triality]].
| | ==Test pages == |
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| ==Outer automorphism groups of complex and real simple Lie algebras==
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| 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 <math>\mathfrak{g}</math>, the automorphism group <math>\operatorname{Aut}(\mathfrak{g})</math> is a [[semidirect product]] of <math>\operatorname{Inn}(\mathfrak{g})</math> and <math>\operatorname{Out}(\mathfrak{g})</math>, i.e., the [[exact sequence|short exact sequence]]
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| : <math>1 \;\xrightarrow{}\; \operatorname{Inn}(\mathfrak{g}) \;\xrightarrow{}\; \operatorname{Aut}(\mathfrak{g}) \;\xrightarrow{}\; \operatorname{Out}(\mathfrak{g}) \;\xrightarrow{}\; 1</math>
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| splits. In the complex simple case, this is a classical result,<ref>{{Harv |Fulton |Harris |1991 |loc = Proposition D.40}}</ref> whereas for real simple Lie algebras, this fact has been proven as recently as 2010.<ref name="JOLT">[http://www.heldermann.de/JLT/JLT20/JLT204/jlt20035.htm]</ref>
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| == Structure ==
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| The [[Schreier conjecture]] asserts that Out(''G'') is always a [[solvable group]] when ''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.
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| == Dual to center ==
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| The outer automorphism group is [[duality (mathematics)|dual]] to the center in the following sense: conjugation by an element of ''G'' is an automorphism, yielding a map <math>\sigma\colon G \to \operatorname{Aut}(G).</math> 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 [[short exact sequence]]:
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| :<math>Z(G) \hookrightarrow G \overset{\sigma}{\to} \operatorname{Aut}(G) \twoheadrightarrow \operatorname{Out}(G).</math>
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| == 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]].
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| === Topology of surfaces ===
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| The outer automorphism group is important in the [[topology]] of [[surface]]s because there is a connection provided by the [[Dehn–Nielsen theorem]]: the extended [[mapping class group]] of the surface is the Out of its [[fundamental group]].
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| ==Puns==
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| The term "outer automorphism" lends itself to [[pun]]s:
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| the term ''outermorphism'' is sometimes used for "outer automorphism",
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| and a particular [[Geometric group action|geometry]] on which <math>\scriptstyle\operatorname{Out}(F_n)</math> acts is called ''[[Out(Fn)#Outer space|outer space]]''.
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| ==External links== | |
| *[http://brauer.maths.qmul.ac.uk/Atlas/v3/ ATLAS of Finite Group Representations-V3]
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| (contains a lot of information on various classes of finite groups (in particular sporadic simple groups), including the order of Out(''G'') for each group listed. | |
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| ==See also==
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| *[[Mapping class group]]
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| *[[Out(Fn)|Out(F<sub>n</sub>)]]
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| ==References==
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| {{Refimprove|date=November 2009}}
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| {{reflist}}
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| {{refbegin}}
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| {{refend}}
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| [[Category:Group theory]]
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| [[Category:Group automorphisms]]
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