# Group isomorphic to its automorphism group

From Groupprops

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

## Contents

## Definition

A group is termed a **group isomorphic to its automorphism group** if it is isomorphic to its automorphism group.

## Relation with other properties

### Stronger properties

### Related properties

## Facts

- A nontrivial group of prime power order cannot be a complete group, because a group of prime power order is either of prime order or has outer automorphism class of same prime order. However, it may still be isomorphic to its automorphism group. The only known example so far is dihedral group:D8. Whether there exist other groups isomorphic to their automorphism groups is an open problem.