Some aspects of marginal automorphisms of a finite p-group

Let be a free group, be a variety of groups defined by the set of laws and be a finite -nilpotent -group. The automorphism of is said to be a marginal automorphism (with respect to ), if for all , , where denotes the marginal subgroup of . An automorphism of is called an IA-automorphism if for each . An automorphism of is called a class preserving if for all , there exists an element such that . Let , and respectively, denote the group of all marginal automorphisms, IA-automorphisms and class preserving automorphisms of . In this paper, first we give a necessary and sufficient condition on a finite -nilpotent -group such that each marginal automorphism of fixes the center of element-wise. Then we characterize all finite -nilpotent -groups such that . Finally, we obtain a necessary and sufficient condition for a finite -nilpotent -group such that .