# 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.

## Particular cases

Group | Order | Automorphism group | Order | Endomorphism structure page | More information |
---|---|---|---|---|---|

trivial group 1 | trivial group | 1 | |||

cyclic group:Z2 | 2 | trivial group | 1 | endomorphism structure of cyclic group:Z2 | For a finite cyclic group of order , the automorphism group is of order where denotes the Euler totient function. Further, the automorphism group is cyclic iff is 2,4, a power of an odd prime, or twice a power of an odd prime. In particular, for a prime , the automorphism group of the cyclic group of order is the cyclic group of order . |

cyclic group:Z3 | 3 | cyclic group:Z2 | 2 | endomorphism structure of cyclic group:Z3 | For a finite cyclic group of order , the automorphism group is of order where denotes the Euler totient function. Further, the automorphism group is cyclic iff is 2,4, a power of an odd prime, or twice a power of an odd prime. In particular, for a prime , the automorphism group of the cyclic group of order is the cyclic group of order . |

cyclic group:Z4 | 4 | cyclic group:Z2 | 2 | endomorphism structure of cyclic group:Z4 | For a finite cyclic group of order , the automorphism group is of order where denotes the Euler totient function. Further, the automorphism group is cyclic iff is 2,4, a power of an odd prime, or twice a power of an odd prime. |

Klein four-group | 4 | symmetric group:S3 | 6 | endomorphism structure of Klein four-group | In general, for an elementary abelian group of order , the automorphism group is the general linear group . |

cyclic group:Z5 | 5 | cyclic group:Z4 | 4 | endomorphism structure of cyclic group:Z5 | For a finite cyclic group of order , the automorphism group is of order where denotes the Euler totient function. Further, the automorphism group is cyclic iff is 2,4, a power of an odd prime, or twice a power of an odd prime. In particular, for a prime , the automorphism group of the cyclic group of order is the cyclic group of order . |

symmetric group:S3 | 6 | symmetric group:S3 | 6 | endomorphism structure of symmetric group:S3 | See symmetric groups are complete and endomorphism structure of symmetric group:S3 |

cyclic group:Z6 | 6 | cyclic group:Z2 | 2 | endomorphism structure of cyclic group:Z6 | For a finite cyclic group of order , the automorphism group is of order where denotes the Euler totient function. Further, the automorphism group is cyclic iff is 2,4, a power of an odd prime, or twice a power of an odd prime. |

cyclic group:Z7 | 7 | cyclic group:Z6 | 6 | endomorphism structure of cyclic group:Z7 | In particular, for a prime , the automorphism group of the cyclic group of order is the cyclic group of order . |

cyclic group:Z8 | 8 | cyclic group:Z4 | 4 | endomorphism structure of cyclic group:Z8 | For a finite cyclic group of order , the automorphism group is of order where denotes the Euler totient function. Further, the automorphism group is cyclic iff is 2,4, a power of an odd prime, or twice a power of an odd prime. |

direct product of Z4 and Z2 | 8 | dihedral group:D8 | 8 | endomorphism structure of direct product of Z4 and Z2 | |

dihedral group:D8 | 8 | dihedral group:D8 | 8 | endomorphism structure of dihedral group:D8 | |

quaternion group | 8 | symmetric group:S4 | 24 | endomorphism structure of quaternion group | |

elementary abelian group:E8 | 8 | projective special linear group:PSL(3,2) | 168 | endomorphism structure of projective special linear group:PSL(3,2) | In general, for an elementary abelian group of order , the automorphism group is the general linear group . In this case, , so we get , which is isomorphic to . |

alternating group:A4 | 12 | symmetric group:S4 | 24 | endomorphism structure of alternating group:A4 | |

symmetric group:S4 | 24 | symmetric group:S4 | 24 | endomorphism structure of symmetric group:S4 | symmetric groups are complete: the symmetric group is a complete group if . |