some aspects of margi al automorphisms of a finite p-group

Let f be a free group, v be a variety of groups defined by the set of laws v ⊆ f and g be a finite v-nilpotent p-group. the automorphism α of g is said to be a marginal automorphism (with respect to v ), if for all x ∈ g, x−¹xα ∈ v ⋆(g), where v ⋆(g) denotes the marginal subgroup of g. an automorphism α of g is called an ia-automorphism if x−¹xα ∈ g′ for each x ∈ g. an automorphism α of g is called a class preserving if for all x ∈ g, there exists an element g ∈ g such that xα = g−¹xg . let autv ⋆ (g), autg′ (g) and autc(g) respectively, denote the group of all marginal automorphisms, ia-automorphisms and class preserving automorphisms of g. in this paper, first we give a necessary and sufficient condition on a finite v-nilpotent p-group g such that each marginal automorphism of g fixes the center of g element-wise. then we characterize all finite v-nilpotent p-groups g such that ⋆ ′ autv (g) = autg (g). finally, we obtain a necessary and sufficient condition for a finite v-nilpotent p-group g such that autv ⋆ (g) = aut (g). c