LAUSR

Let G be a finite non-abelian p-group, where p is a prime. An automorphism $$\alpha $$ of G is called an nth class-preserving if for each $$x\in G$$ , there exists an element $$g_x\in \gamma _n(G)$$ such that $$\alpha (x)=g_x^{-1}xg_x$$ . An automorphism $$\alpha $$ of G is called a central automorphism if $$x^{-1}\alpha (x)\in Z(G)$$ for all $$x\in G$$ . Let $${{\,\mathrm{Aut}\,}}_{c}^n(G)$$ and $${{\,\mathrm{Autcent}\,}}(G)$$ , 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 $${{\,\mathrm{Aut}\,}}_{c}^n(G)={{\,\mathrm{Autcent}\,}}(G)$$ .