Perfect groups and normal subgroups related to an automorphism

Abstract

In this paper, we introduce the concept of αα -normal subgroup of a finite group G, where αα is an automorphism of G. We also introduce the concept of absolute normal subgroup and investigate all absolute normal subgroups of some groups. Furthermore, we define a group G αα -perfect if the αα -commutator subgroup of G coincides with G. We prove that for every finite abelian group G, there exists a finite abelian group H and α∈Aut(H)α∈Aut(H) such that Dα(H)=GDα(H)=G .