On Marginal Automorphisms of a Group Fixing the Certain Subgroup

Let W be a variety of groups defined by a set W of laws and G be a finite p-group in W. The automorphism α of a group G is said to bea marginal automorphism (with respect to W), if for all x ∈ G, x−1α(x) ∈ W∗(G), where W∗(G) is the marginal subgroup of G. Let M,N be two normalsubgroups of G. By AutM(G), we mean the subgroup of Aut(G) consistingof all automorphisms which centralize G/M. AutN(G) is used to show thesubgroup of Aut(G) consisting of all automorphisms which centralize N. We denote AutN(G)∩AutM(G) by AutMN (G). In this paper, we obtain a necessary and sufficient condition that Autw∗ (G) = AutW∗(G)W∗(G)(G).