Year 2020, Volume 49 , Issue 1, Pages 273 - 281 2020-02-06

Acentralizers of Abelian groups of rank 2

Zahar MOZAFAR [1] , Bijan TAERİ [2]


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.
Automorphism, centralizer, Acentralizer, finite groups
  • [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.
Primary Language en
Subjects Mathematics
Journal Section Mathematics
Authors

Orcid: 0000-0002-1039-7975
Author: Zahar MOZAFAR
Institution: Isfahan University of Technology
Country: Iran


Orcid: 0000-0001-7345-1281
Author: Bijan TAERİ
Institution: Isfahan University of Technology
Country: Iran


Dates

Publication Date : February 6, 2020

Bibtex @research article { hujms546988, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe University}, year = {2020}, volume = {49}, pages = {273 - 281}, doi = {10.15672/hujms.546988}, title = {Acentralizers of Abelian groups of rank 2}, key = {cite}, author = {MOZAFAR, Zahar and TAERİ, Bijan} }
APA MOZAFAR, Z , TAERİ, B . (2020). Acentralizers of Abelian groups of rank 2. Hacettepe Journal of Mathematics and Statistics , 49 (1) , 273-281 . DOI: 10.15672/hujms.546988
MLA MOZAFAR, Z , TAERİ, B . "Acentralizers of Abelian groups of rank 2". Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 273-281 <https://dergipark.org.tr/en/pub/hujms/issue/52287/546988>
Chicago MOZAFAR, Z , TAERİ, B . "Acentralizers of Abelian groups of rank 2". Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 273-281
RIS TY - JOUR T1 - Acentralizers of Abelian groups of rank 2 AU - Zahar MOZAFAR , Bijan TAERİ Y1 - 2020 PY - 2020 N1 - doi: 10.15672/hujms.546988 DO - 10.15672/hujms.546988 T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 273 EP - 281 VL - 49 IS - 1 SN - 2651-477X-2651-477X M3 - doi: 10.15672/hujms.546988 UR - https://doi.org/10.15672/hujms.546988 Y2 - 2018 ER -
EndNote %0 Hacettepe Journal of Mathematics and Statistics Acentralizers of Abelian groups of rank 2 %A Zahar MOZAFAR , Bijan TAERİ %T Acentralizers of Abelian groups of rank 2 %D 2020 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 49 %N 1 %R doi: 10.15672/hujms.546988 %U 10.15672/hujms.546988
ISNAD MOZAFAR, Zahar , TAERİ, Bijan . "Acentralizers of Abelian groups of rank 2". Hacettepe Journal of Mathematics and Statistics 49 / 1 (February 2020): 273-281 . https://doi.org/10.15672/hujms.546988
AMA MOZAFAR Z , TAERİ B . Acentralizers of Abelian groups of rank 2. Hacettepe Journal of Mathematics and Statistics. 2020; 49(1): 273-281.
Vancouver MOZAFAR Z , TAERİ B . Acentralizers of Abelian groups of rank 2. Hacettepe Journal of Mathematics and Statistics. 2020; 49(1): 281-273.