Difference between revisions of "Automorphism group of a group"

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===Definition with symbols===
 
===Definition with symbols===
  
The '''automorphism group''' of a [[group]] <math>G</math>, denoted <math>Aut(G)</math>, is a set whose elements are automorphisms <math>\sigma:G \to G</math>, and where the group multiplication is composition of automorphisms. In other words, its group structure is obtained as a subgroup of <math>\operatorname{Sym}(G)</math>, the group of all permutations on <math>G</math>.
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The '''automorphism group''' of a [[group]] <math>G</math>, denoted <math>\operatorname{Aut}(G)</math>, is a set whose elements are automorphisms <math>\sigma:G \to G</math>, and where the group multiplication is composition of automorphisms. In other words, its group structure is obtained as a subgroup of <math>\operatorname{Sym}(G)</math>, the group of all permutations on <math>G</math>.
  
 
==Subgroups==
 
==Subgroups==
  
Every [[group-closed automorphism property]] gives rise to a [[normal subgroup]] of the automorphism group. Examples are the property of being an [[inner automorphism]], [[class automorphism]], [[extensible automorphism]].
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Every [[group-closed automorphism property]] gives rise to a [[normal subgroup]] of the automorphism group. Some of the most important examples are given below:
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{| class="sortable" border="1"
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! Group-closed automorphism property !! Meaning !! Corresponding normal subgroup of the automorphism group
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|-
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| [[inner automorphism]] || can be expressed as [[conjugation]] by an element of the group, i.e., there exists <math>g \in G</math> such that the map has the form <math>x \mapsto gxg^{-1}</math> || it is called the [[inner automorphism group]] and is isomorphic to the [[quotient group]] <math>G/Z(G)</math> where <math>Z(G)</math> is the [[center]]. See [[group acts as automorphisms by conjugation]].
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|-
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| [[class-preserving automorphism]] || sends every element to within its automorphism class || the class-preserving automorphism group
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|-
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| [[IA-automorphism]] || sends every coset of the [[derived subgroup]] to itself, or equivalently, induces the identity map on the [[abelianization]]. || the IA-automorphism group
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|-
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| [[center-fixing automorphism]] || fixes every element of the center || the center-fixing automorphism group
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|-
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| [[monomial automorphism]] || can be expressed using a monomial formula || the momomial automorphism group
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|-
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| [[normal automorphism]] || sends every normal subgroup to itself || the normal automorphism group
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|}
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==Facts==
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* [[Extensible equals inner]]: An automorphism of a group has the property that it can be extended to an automorphism for any bigger group containing it if and only if the automorphism is an [[inner automorphism]].
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* [[Quotient-pullbackable equals inner]]: An automorphism of a group has the property that it can be pulled back to an automorphism for any group admitting it as a quotient, if and only if the automorphism is an inner automorphism.

Revision as of 03:08, 28 May 2013

This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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Definition

Symbol-free definition

The automorphism group of a group is defined as a group whose elements are all the automorphisms of the base group, and where the group operation is composition of automorphisms. In other words, it gets a group structure as a subgroup of the group of all permutations of the group.

Definition with symbols

The automorphism group of a group G, denoted \operatorname{Aut}(G), is a set whose elements are automorphisms \sigma:G \to G, and where the group multiplication is composition of automorphisms. In other words, its group structure is obtained as a subgroup of \operatorname{Sym}(G), the group of all permutations on G.

Subgroups

Every group-closed automorphism property gives rise to a normal subgroup of the automorphism group. Some of the most important examples are given below:

Group-closed automorphism property Meaning Corresponding normal subgroup of the automorphism group
inner automorphism can be expressed as conjugation by an element of the group, i.e., there exists g \in G such that the map has the form x \mapsto gxg^{-1} it is called the inner automorphism group and is isomorphic to the quotient group G/Z(G) where Z(G) is the center. See group acts as automorphisms by conjugation.
class-preserving automorphism sends every element to within its automorphism class the class-preserving automorphism group
IA-automorphism sends every coset of the derived subgroup to itself, or equivalently, induces the identity map on the abelianization. the IA-automorphism group
center-fixing automorphism fixes every element of the center the center-fixing automorphism group
monomial automorphism can be expressed using a monomial formula the momomial automorphism group
normal automorphism sends every normal subgroup to itself the normal automorphism group

Facts

  • Extensible equals inner: An automorphism of a group has the property that it can be extended to an automorphism for any bigger group containing it if and only if the automorphism is an inner automorphism.
  • Quotient-pullbackable equals inner: An automorphism of a group has the property that it can be pulled back to an automorphism for any group admitting it as a quotient, if and only if the automorphism is an inner automorphism.