Some aspects of marginal automorphisms of a finite $p$-group

Document Type : Research Paper


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



Let $F$ be a free group, $\mathcal{V}$ be a variety of groups defined by the set of laws $V\subseteq F$ and $G$ be a finite $\mathcal{V}$-nilpotent $p$-group. The automorphism $\alpha$ of $G$ is said to be a marginal automorphism (with respect to $V$), if for all $x\in G$, $x^{-1}x^{\alpha}\in V^{\star}(G)$, where $V^{\star}(G)$ denotes the marginal subgroup of $G$. An automorphism $\alpha$ of $G$ is called an IA-automorphism if $x^{-1}x^{\alpha}\in G'$ for each $x\in G$. An automorphism $\alpha$ of $G$ is called a class preserving if for all $x\in G$, there exists an element $g_x\in G$ such that $x^{\alpha}=g_x^{-1}xg_x$. Let $\operatorname{Aut}^{V^{\star}}(G)$, $\operatorname{Aut}^{G'}(G)$ and $\operatorname{Aut}_c(G)$ 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 $\mathcal{V}$-nilpotent $p$-group $G$ such that each marginal automorphism of $G$ fixes the center of $G$ element-wise. Then we characterize all finite $\mathcal{V}$-nilpotent $p$-groups $G$ such that $\operatorname{Aut}^{V^{\star}}(G)=\operatorname{Aut}^{G'}(G)$. Finally, we obtain a necessary and sufficient condition for a finite $\mathcal{V}$-nilpotent $p$-group $G$ such that $\operatorname{Aut}^{V^{\star}}(G)=\operatorname{Aut}_c(G)$.


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