Acentralizers of Abelian groups of rank 2

Authors : Zahar MOZAFAR, Bijan TAERİ

Pages : 273-281

Doi:10.15672/hujms.546988

View : 12 | Download : 6

Publication Date : 2020-02-06

Article Type : Research Paper

Abstract :Let $G$ be a group. The Acentralizer of an automorphism $\alpha$ of $G$, is the subgroup of fixed points of $\alpha$, i.e.,  $C_Ginsert ignore into journalissuearticles values(\alpha);= \{g\in G \mid \alphainsert ignore into journalissuearticles values(g);=g\}$. 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 $C_Ginsert ignore into journalissuearticles values(\alpha);=U$. Also we find the Acentralizers of infinite two-generator Abelian groups.
Keywords : Automorphism, centralizer, Acentralizer, finite groups