Some aspects of marginal automorphisms of a finite p-group

Document Type : Research Paper

Author

Department of Mathematics, Payame Noor University (PNU), 19395-3697, Tehran, Iran.

Abstract

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 gx∈G such that xα=gx−1xgx. 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)=Autc(G).

Keywords

References

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Algebraic Structures and Their Applications
Volume 9, Issue 1
February 2022
Pages 133-143

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