# Locally inner automorphism

## Contents

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 $\sigma$ of a group $G$ is termed a locally inner automorphism if, for any elements $x_1,x_2,\dots,x_n \in G$, there exists $g \in G$ such that $gx_ig^{-1} = \sigma(x_i)$ for $1 \le i \le n$.

## 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