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 \(n \geq 2\), 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 \(\phi(Aut_{c}^{n-1}(G))=Aut_{c}^{n-1}(H)\). Also, for a finite non-abelian \(p\)-group \(G\) satisfying a certain natural hypothesis, we give some necessary and sufficient conditions for \(Autcent(G) = Aut_c^{n-1}(G)\). Furthermore, for a finite non-abelian \(p\)-group \(G\) we study the equality of \(Autcent(G)\) with \(Aut_{Z(G)}^{\gamma_{n}(G)}(G)\).