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 properties
VIEW 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.
