- Hacettepe Journal of Mathematics and Statistics
- Vol: 49 Issue: 1
- Acentralizers of Abelian groups of rank 2
Acentralizers of Abelian groups of rank 2
Authors : Zahar Mozafar, Bijan Taeri
Pages : 273-281
Doi:10.15672/hujms.546988
View : 10 | Download : 11
Publication Date : 2020-02-06
Article Type : Research
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