Group isomorphic to its automorphism group

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 automorphism group if it is isomorphic to its automorphism group.

Relation with other properties

Stronger properties

• Complete group

Related properties

• Centerless group
• Group isomorphic to its inner automorphism group

Examples

The examples for finite groups of order less than or equal to 12 are trivial group, symmetric group:S3, dihedral group:D8 and dihedral group:D12.

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.