# Group isomorphic to its inner automorphism group

## 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 is termed a group isomorphic to its inner automorphism group if the group is isomorphic to its inner automorphism group, or equivalently, the group is isomorphic to the quotient by its center.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Centerless group center is trivial isomorphic to inner automorphism group not implies centerless |FULL LIST, MORE INFO

## Facts

For a finite group, and more generally for a Hopfian group, being isomorphic to its inner automorphism group is equivalent to being centerless.