On Equality of Certain Automorphism Groups

Surjeet Kour, Vishakha

Published 2015-05-21Version 1

Let $G = H\times A$ be a finite group, where $H$ is a purely non-abelian subgroup of $G$ and $A$ is a non-trivial abelian factor of $G$. Then, for $n \geq 2$, we show that there exists an isomorphism $\phi : Aut_{Z(G)}^{\gamma_{n}(G)}(G) \rightarrow Aut_{Z(H)}^{\gamma_{n}(H)}(H)$ such that $\phi(Aut_{c}^{n-1}(G))=Aut_{c}^{n-1}(H)$. We also give some necessary and sufficient conditions on a finite $p$-group $G$ such that $Autcent(G)=Aut_{c}^{n-1}(G)$ . Furthermore, for a finite non-abelian $p$-group $G$, we give some necessary and sufficient conditions for $Aut_{Z(G)}^{\gamma_{n}(G)}(G)$ to be equal to $Aut_{\gamma_2(G)}^{Z(G)}(G)$.