KaydedilenlerDerginizi ekleyin

Dergiler / Hacettepe Journal of Mathematics and Statistics / 2026 / Volume: 49 Issue: 1

Acentralizers of Abelian groups of rank 2

Zahar Mozafar(Isfahan University of Technology)0000-0002-1039-7975
Bijan Taeri(Isfahan University of Technology)0000-0001-7345-1281
Sayfa
—
Yayın Tarihi
2020-02-06
🏛️Bu makalenin bibliyografik verileri ve tam metin bağlantısı DergiPark üzerinden sağlanmıştır.
DergiPark Kaydı ↗
✨

Yapay Zeka Hızlı Bakış & Çıkarımlar

AI Insight
⚡ TL;DR(1 Cümlede Hızlı Çıkarım)

Also we find the Acentralizers of infinite two-generator Abelian groups.

📌 3 Maddede Bu Makale

🎯
Amaç & Hipotez:The Acentralizer of an automorphism $\alpha$ of $G$, is the subgroup of fixed points of $\alpha$, i.e., $C_G(\alpha)= \{g\in G \mid \alpha(g)=g\}$.
📊
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 $G$.
💡
Sonuç & Katkı:Also we find the Acentralizers of infinite two-generator Abelian groups.
🔬Tür:Research Article
📍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.

🔖

Bu Makaleye Atıf Yap

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

📄 Tam Metin PDF Okuyucu

📄

PDF görüntüleyici hazırlanıyor...

Kaynakça (15)

  1. A. Abdollahi, S.M.J. Amiri, and A.M. Hassanabadi, Groups with specific number of centralizers, Houston J. Math. 33 (1), 43–57, 2007.
  2. S.M.J. Amiri and H. Rostami, Centralizers and the maximum size of the pairwise noncommuting elements in finite groups, Hacet. J. Math Stat. 46 (2), 193–198, 2017.
  3. A.R. Ashrafi, On finite groups with a given number of centralizers, Algebra Colloq. 7 (2), 139–146, 2000.
  4. A.R. Ashrafi, Counting the centralizers of some finite groups, J. Korean Comput. Appl. Math. 7 (1), 115–124, 2000.
  5. A.R. Ashrafi and B. Taeri, On finite groups with a certain number of centralizers, J. Appl. Math. Comput. 17 (12), 217–227, 2005.
  6. A.R. Ashrafi and B. Taeri, On finite groups with exactly seven element centralizers, J. Appl. Math. Comput. 22 (1-2), 403–410, 2006.
  7. S.M. Belcastro and G.J. Sherman, Counting centralizers in finite groups, Math. Mag. 67 (5), 366–374, 1994.
  8. Gr.G. Călugăreanu, The total number of subgroups of a finite Abelian group, Sci. Math. Jpn. 60 (1), 157–167, 2004.
  9. J. Dutta, A characterization of 4-centralizer groups, Chin. J. Math. (N.Y.) Article ID:871072, 2 pages, 2013.
  10. J. Dutta, On a problem posed by Belcastro and Sherman, Kyungpook Math. J. 56 (1), 121–123, 2016.
  11. M. Golasiński and D.L. Goncalves, On automorphisms of finite Abelian p-groups, Math. Slovaca 58 (4), 405–412, 2008.
  12. M.M. Nasrabadi and A. Gholamian, On finite n-Acentralizers groups, Comm. Algebra 43 (2), 378–383, 2015.
  13. R.K. Nath, Commutativity degree of a class of finite groups and consequences, Bull. Aust. Math. Soc. 88 (3), 448–452, 2013.
  14. R. Schmidt, Subgroup lattices of groups, De Gruyter Exp. Math., 14. Walter de Gruyter, 1994.
  15. M. Zarrin, On element-centralizers in finite groups, Arch. Math. (Basel) 93, 497–503, 2009.