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

Abstract

Let G be a finite group and M, N be two normal subgroups of G. Let Aut M N (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 nisoclinic groups, then there exists an isomorphism from AutZn(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 of class at most (n + 1), with some suitable conditions on γn+1(G), we prove that AutZn(G) (G) is isomorphic to the group of inner automorphisms of a quotient group of G.