On nilpotency of the group of outer class-preserving automorphisms of a group

Abstract

Let G be a group and Zj(G), for j ≥ 0, be the jth term in the upper central series of G. We prove that if Outc(G/Zj(G)), the group of outer class-preserving automorphisms of G/Zj(G), is nilpotent of class k, then Outc(G) is nilpotent of class at most j + k. Moreover, if Outc(G/Zj(G)) is a trivial group, then Outc(G) is nilpotent of class at most j. As an application we prove that if γi(G)/γi(G) ∩ Zj(G) is cyclic then Outc(G) is nilpotent of class at most i + j, where γi(G), for i ≥ 1, denotes the ith term in the lower central series of G. This extends an earlier work of the author, where this assertion was proved for j = 0. We also improve bound on the nilpotency class of Outc(G) for some classes of nilpotent groups G.