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AI Summary & Key Takeaways
AI Insight⚡ TL;DR(Too Long; Didn't Read)
Also we find the Acentralizers of infinite two-generator Abelian groups.
📌 3 Key Takeaways
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Objective & Hypothesis:The Acentralizer of an automorphism $\alpha$ of $G$, is the subgroup of fixed points of $\alpha$, i.e., $C_G(\alpha)= \{g\in G \mid \alpha(g)=g\}$.
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Key Findings & Data: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 $G$.
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Conclusion & Impact:Also we find the Acentralizers of infinite two-generator Abelian groups.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Automorphism · centralizer · Acentralizer
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.
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Cite This Article
Mozafar, Z., Taeri, B. (2020). Acentralizers of Abelian groups of rank 2. Hacettepe Journal of Mathematics and Statistics, Volume: 49 Issue: 1. https://doi.org/10.15672/hujms.546988