# Group in which every locally inner automorphism is inner

## Contents

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Definition

A group $G$ is termed a group in which every locally inner automorphism is inner if every locally inner automorphism of $G$ is an inner automorphism of $G$. Here, locally inner means that for any finite subset, its action on that finite subset agrees with some inner automorphism.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
group with finitely generated inner automorphism group the inner automorphism group is a finitely generated group finitely generated inner automorphism group implies every locally inner automorphism is inner |FULL LIST, MORE INFO
FZ-group the inner automorphism group is finite (via finitely generated inner automorphism group) Template:Intermediate notions sohrt
finitely generated group the whole group is finitely generated, i.e., has a finite generating set (via finitely generated inner automorphism group) Group with finitely generated inner automorphism group|FULL LIST, MORE INFO
finite group the whole group is finite (via finitely generated, also via FZ) Group with finitely generated inner automorphism group|FULL LIST, MORE INFO
abelian group the whole group is abelian, so the inner automorphism group is trivial (via FZ) Group with finitely generated inner automorphism group|FULL LIST, MORE INFO