On Equality of Certain Automorphism Groups

Abstract

Let $G=H\times A$ be a group, where $H$ is a purely non-abelian subgroup of $G$ and $A$ is a non-trivial abelian factor of $G$. Then, for $n\ge 2$, we show that there exists an isomorphism $\varphi :Au{t}_{Z(G)}^{{\gamma }_n(G)}(G)\to Au{t}_{Z(H)}^{{\gamma }_n(H)}(H)$ such that $\varphi (Au{t}_c^{n-1}(G))=Au{t}_c^{n-1}(H)$. Also, for a finite non-abelian $p$-group $G$ satisfying a certain natural hypothesis, we give some necessary and sufficient conditions for $Autcent(G)=Au{t}_c^{n-1}(G)$. Furthermore, for a finite non-abelian $p$-group $G$ we study the equality of $Autcent(G)$ with $Au{t}_{Z(G)}^{{\gamma }_n(G)}(G)$.

Accepted in Communications in Algebra