# Group isomorphic to its inner automorphism group

From Groupprops

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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 propertiesVIEW 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 |

### Related properties

## Facts

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