# 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>. | + | 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. | + | Every [[group-closed automorphism property]] gives rise to a [[normal subgroup]] of the automorphism group. Some of the most important examples are given below: |

+ | |||

+ | {| class="sortable" border="1" | ||

+ | ! 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 <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]]. | ||

+ | |- | ||

+ | | [[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. |

## Revision as of 03:08, 28 May 2013

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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 , denoted , is a set whose elements are automorphisms , and where the group multiplication is composition of automorphisms. In other words, its group structure is obtained as a subgroup of , the group of all permutations on .

## 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 such that the map has the form | it is called the inner automorphism group and is isomorphic to the quotient group where 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.