Group isomorphic to its automorphism group

Definition
A group is termed a group isomorphic to its automorphism group if it is isomorphic to its defining ingredient::automorphism group.

Stronger properties

 * Weaker than::Complete group

Related properties

 * Centerless group
 * Group isomorphic to its inner automorphism group

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.