On ${{\varvec{n}}}$-th class preserving automorphisms of ${{\varvec{n}}}$-isoclinism family

Abstract

Let G be a finite group and let M and N be two normal subgroups of G. Let $\hbox {Aut}_N^M(G)$ denote the group of all automorphisms of G which fix N element-wise and act trivially on G / M. Let n be a positive integer. In this article, we have shown that if G and H are two n-isoclinic groups, then there exists an isomorphism from $\hbox {Aut}_{Z_n(G)}^{\gamma _{n+1}(G)}(G)$ to $\hbox {Aut}_{Z_n(H)}^{\gamma _{n+1}(H)}(H)$, which maps the group of n-th class preserving automorphisms of G to the group of n-th class preserving automorphisms of H. Also, for a nilpotent group G of class $(n+1)$, if $\gamma _{n+1}(G)$ is cyclic, then we prove that $\hbox {Aut}_{Z_n(G)}^{\gamma _{n+1}(G)}(G)$ is isomorphic to the group of inner automorphisms of a quotient group of G.