Perfect groups and normal subgroups related to an automorphism

Abstract

In this paper, we introduce the concept of $\alpha$-normal subgroup of a finite group G, where $\alpha$ 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 $\alpha$-perfect if the $\alpha$-commutator subgroup of G coincides with G. We prove that for every finite abelian group G, there exists a finite abelian group H and $\alpha \in Aut(H)$ such that $D_{\alpha }(H)=G$.