A note on marginal and central automorphisms of a finite p-group

Let [Formula: see text] be a free group, and [Formula: see text] be a variety of groups defined by the set of laws [Formula: see text]. The automorphism [Formula: see text] of a group [Formula: see text] is called a marginal automorphism (with respect to [Formula: see text]), if it induces the identity automorphism on [Formula: see text], where [Formula: see text] denotes the marginal subgroup of [Formula: see text]. Let [Formula: see text] denote the group of all marginal automorphisms of [Formula: see text]. An automorphism [Formula: see text] of a group [Formula: see text] is called a central automorphism, if [Formula: see text] induces the identity on [Formula: see text]. In this paper, we find necessary and sufficient condition on a finite [Formula: see text]-nilpotent [Formula: see text]-group such that every marginal automorphism is inner. Also, we obtain a necessary and sufficient condition for a finite [Formula: see text]-nilpotent [Formula: see text]-group [Formula: see text] such that [Formula: see text], where [Formula: see text] is the group of all inner automorphisms of [Formula: see text]. Finally, by considering specific marginal subgroups, we list some applications of the previous results on equalities of central automorphisms.