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Dergiler / Hacettepe Journal of Mathematics and Statistics / 2020 / Cilt: 49 - Sayı: 1

Acentralizers of Abelian groups of rank 2

Zahar MOZAFAR
Bijan TAERİ
Sayfa
273–281
Yayın Tarihi
2022-10-18
DOI
—
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Also we find the Acentralizers of infinite two-generator Abelian groups.

📌 3 Maddede Bu Makale

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Amaç & Hipotez: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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Temel Bulgular & Veri: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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Sonuç & Katkı:Also we find the Acentralizers of infinite two-generator Abelian groups.
🔬Tür:Araştırma Makalesi
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler: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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Bu Makaleye Atıf Yap

MOZAFAR, Z., TAERİ, B. (2022). Acentralizers of Abelian groups of rank 2. Hacettepe Journal of Mathematics and Statistics, Cilt: 49 - Sayı: 1, 273–281. https://ajindex.com/articles/529581

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