On nth class preserving automorphisms of n-isoclinism family

Abstract

Let G be a finite group and M,N be two normal subgroups of G. Let AutMN(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 Autγn+1(G)Zn(G)(G) to Autγn+1(H)Zn(H)(H), which maps the group of nth class preserving automorphisms of G to the group of nth class preserving automorphisms of H. Also, for a nilpotent group of class at most (n+1), with some suitable conditions on γn+1(G), we prove that Autγn+1(G)Zn(G)(G) is isomorphic to the group of inner automorphisms of a quotient group of G.