Finite complete group

This article defines a property that can be evaluated for finite groups (and hence, a particular kind of group property)
View other properties of finite groups OR View all group properties

Definition

A finite complete group is a finite group that is also a complete group: it satisfies the following two conditions:

• It is a centerless group: its center is trivial.
• It is a group in which every automorphism is inner.

In other words, a finite complete group is a finite group $G$ such that the natural map $G \to \operatorname{Aut}(G)$ is an isomorphism.

Relation with other properties

Stronger properties

• Automorphism group of finite simple non-abelian group
• Automorphism group of finite characteristically simply non-abelian group: For full proof, refer: Characteristically simple and non-abelian implies automorphism group is complete
• Finite complete solvable group