Acentralizers of Abelian groups of rank 2

Abstract

Let $G$ be a group. The Acentralizer of an automorphism $\alpha$ of $G$, is the subgroup of fixed points of $\alpha$, i.e.,  $C_G(\alpha)= \{g\in G \mid \alpha(g)=g\}$. We show that if $G$ is a  finite  Abelian  $p$-group of rank $2$, where $p$ is an odd prime, then the number of Acentralizers of $G$ is exactly the number of subgroups of $G$. More precisely, we show that for each  subgroup $U$ of $G$, there exists an automorphism $\alpha$ of $G$ such that $C_G(\alpha)=U$. Also we find the Acentralizers of infinite two-generator Abelian groups.

Keywords

• Automorphism,centralizer,Acentralizer,finite groups

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