On Finite p-Groups Whose Central Automorphisms Are All nth Class-Preserving

Let G be a finite non-abelian p-group, where p is a prime. An automorphism a of G is called an nth class-preserving if for each x. G, there exists an element gx..n(G) such that a(x) = g-1 x xgx. An automorphism a of G is called a central automorphism if x-1a( x). Z(G) for all x. G. Let Autnc (G) and Autcent(G), respectively, denote the group of all nth class-preserving and central automorphisms of G. We give necessary and sufficient conditions for a finite p-group G of class n+ 1 such that Autnc (G) = Autcent(G).