# Automorphism group of a group

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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.