# Locally inner automorphism

From Groupprops

BEWARE!This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

This article defines an automorphism property, viz a property of group automorphisms. Hence, it also defines a function property (property of functions from a group to itself)

View other automorphism properties OR View other function properties

## Definition

QUICK PHRASES: inner on finite tuples, inner on finitely generated subgroups

### Definition with symbols

An automorphism of a group is termed a **locally inner automorphism** if, for any elements , there exists such that for .

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Inner automorphism |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Class-preserving automorphism | sends every element to within its conjugacy class | locally inner implies class-preserving | class-preserving not implies locally inner | |FULL LIST, MORE INFO |

Center-fixing automorphism | fixes every element of the center | (via class-preserving) | (via class-preserving) | Class-preserving automorphism|FULL LIST, MORE INFO |

IA-automorphism | induces identity map on the abelianization | (via class-preserving) | (via class-preserving) | Automorphism that preserves conjugacy classes for a generating set, Class-preserving automorphism|FULL LIST, MORE INFO |

Normal automorphism | sends every normal subgroup to itself | Class-preserving automorphism|FULL LIST, MORE INFO | ||

Weakly normal automorphism | sends every normal subgroup to a subgroup of itself | Class-preserving automorphism|FULL LIST, MORE INFO |