Acentralizers of some Finite Groups
Download PDF
Authors: Z. MOZAFAR AND B. TAERI
DOI: 10.46793/KgJMat2502.223M
Abstract:
Let G be a finite group. The acentralizer of an automorphism α of G, is the subgroup of fixed points of α, i.e., CG(α) = {g ∈ G∣α(g) = g}. In this paper we determine the acentralizers of the dihedral group of order 2n, the dicyclic group of order 4n and the symmetric group on n letters. As a result we see that if n ≥ 3, then the number of acentralizers of the dihedral group and the dicyclic group of order 4n are equal. Also we determine the acentralizers of groups of orders pq and pqr, where p, q and r are distinct primes.
Keywords:
Automorphism, centralizer, acentralizer, finite groups.
References:
[1] A. Abdollahi, S. M. J. Amiri and A. M. Hassanabadi, Groups with specific number of centralizers, Houston J. Math. 33(1) (2007), 43–57. https://doi.org/10.1.1.640.5229
[2] A. R. Ashrafi, On finite groups with a given number of centralizers, Algebra Colloq. 7(1) (2000), 139–146. https://doi.org/10.1007/s10011-000-0139-5
[3] A. R. Ashrafi, Counting the centralizers of some finite groups, The Korean Journal of Computational & Applied Mathematics 7(1) (2000), 115–124. https://doi.org/10.1007/BF03009931
[4] A. R. Ashrafi and B. Taeri, On finite groups with a certain number of centralizers, The Korean Journal of Computational & Applied Mathematics 17(1–2) (2005), 217–227. https://doi.org/10.1007/BF02936050
[5] A. R. Ashrafi and B. Taeri, On finite groups with exactly seven element centralizers, The Korean Journal of Computational & Applied Mathematics 22(1–2) (2006), 403–410. https://doi.org/10.1007/BF02896488
[6] S. M. Belcastro and G. J. Sherman, Counting centralizers in finite groups, Math. Mag. 67(5) (1994), 366–374. https://doi.org/10.1080/0025570X.1994.11996252
[7] S. J. Baishya Counting centralizers and z-classes of some F-groups, Comm. Algebra (2021) (to appear). https://doi.org/10.1080/00927872.2021.2008413
[8] Z. Foruzanfar, Z. Mostaghim and M. Rezaei, Classification of finite simple groups according to the number of centralizers, Malays. J. Math. Sci. 12(2) (2018), 211–222.
[9] The GAP Group. Groups, Algorithms and Programing, Version 4.4, 2005. http://www.gap-system.org
[10] M. Ghorbani and F. Nowroozi Larki, Automorphism group of groups of order pqr, Algebraic Structures and their Applications 1(1) (2014), 49–56.
[11] H. Hölder, Die Gruppen der Ordnungen p3, pq2, pqr, p4, Math. Ann. 43 (1893), 371–410. https://doi.org/10.1007/BF01443651
[12] M. A. Iranmanesh and M. H. Zareian, On n-centralizer CA-groups, Comm. Algebra 49(10) (2021), 4186–4195. https://doi.org/10.1080/00927872.2021.1916513
[13] Z. Mozafar and B. Taeri, Acentralizers of abelian groups of rank 2, Hacet. J. Math. Stat. 49(1) (2020), 273–281. https://doi.org/10.15672/hujms.546988
[14] M. M. Nasrabadi and A. Gholamian, On finite n-Acentralizers groups, Comm. Algebra 43(2) (2015), 378–383. https://doi.org/10.1080/00927872.2013.842244
[15] K. H. Rosen, Elementary Number Theory and its Applications, Addison-Wesley, Reading, Massachusetts Menlo Park, California, 1983.
[16] P. Seifizadeh, A. Gholamian, A. Farokhianee and M. A. Nasrabadi, On the autocentralizer subgroups of finite p-groups, Turkish J. Math. 44 (2020), 1802–1812. https://doi.org/10.3906/mat-1911-89
[17] M. Suzuki, Group Theory I, Springer-Verlag, Berlin, Heidelberg, New York, 1982.
[18] M. Zarrin, On element-centralizers in finite groups, Arch. Math. 93 (2009), 497–503. https://doi.org/10.1007/s00013-009-0060-1
Login:
