On central automorphisms that fix the centre elementwise

Abstract

Let G be a group and let Aut c (G) be the group of central automorphisms of G. Let $${{C_{{\rm Aut}_{c}(G)}}(Z(G))}$$ be the set of all central automorphisms of G fixing Z(G) elementwise. In this paper we prove that if G is a finite p-group, then $${C_{{\rm Aut}_{c}(G)}}(Z(G))$$ = Inn(G) if and only if G is abelian or G is nilpotent of class 2 and Z(G) is cyclic.