ON FINITE GROUPS WITH SPECIFIC NUMBER OF CENTRALIZERS

Year 2013, Volume: 13 Issue: 13, 53 - 62, 01.06.2013

Sekhar Jyoti Baishya

Abstract

For a group G, | Cent(G) | denotes the number of distinct centralizers
of its elements. A group G is called n-centralizer if | Cent(G) |= n,
and primitive n-centralizer if | Cent(G) |=| Cent(GZ(G)) |= n. In this paper,
among other things, we investigate the structure of finite groups of odd order
with | Cent(G) |= 9 and prove that if |G| is odd, then | Cent(G) |= 9 if and
only if GZ(G)∼= C7 o C3 or C7 × C7.

Keywords

finite group, n-centralizer group, primitive n-centralizer group

References

• A. Abdollahi, S. M. J. Amiri and A. M. Hassanabadi, Groups with specific number of centralizers, Houston J. Math., 33(1) (2007), 43–57.
• A. R. Ashrafi, On finite groups with a given number of centralizers, Algebra Colloq., 7(2) (2000), 139–146.
• A. R. Ashrafi, Counting the centralizers of some finite groups, Korean J. Com- put. Appl. Math., 7(1) (2000), 115–124.
• A. R. Ashrafi and B. Taeri, On finite groups with exactly seven element cen- tralizers, J. Appl. Math. Comput., 22(1-2) (2006), 403–410.
• A. R. Ashrafi and B. Taeri, On finite groups with a certain number of central- izers, J. Appl. Math. Comput., 17(1-2) (2005), 217–227.
• S. M. Belcastro and G. J. Sherman, Counting centralizers in finite groups, Math. Magazine, 67(5) (1994), 366–374.
• Y. G. Berkovich and E. M. Zhmu´d, Characters of Finite Groups, Part 1, Transl. Math. Monographs 172, Amer. Math. Soc., Providence. RI, 1998.
• A. Castelaz, Commutativity degree of finite groups, MSc. Thesis, Wake Forest University, North Carolina, May, 2010.
• J. H. E. Cohn, On n-sum groups, Math. Scand., 75 (1994), 44–58.
• J. D. Dixon, Problems in Group Theory, Dover Publications, Inc., Mineola, New York, 2007.
• S. Dolfi, M. Herzog and E. Jabara, Finite groups whose noncentral commuting elements have centralizers of equal size, Bull. Aust. Math Soc., 82 (2010), 293–
• H. Kurzweil and B. Stellmacher, The Theory of Finite Groups, An Introduc- tion, Springer-Verlag, New York, Inc. 2004.
• D. J. Rusin, What is the probability that two elements of a finite group com- mute?, Pacific J. Math., 82(1) (1979), 237–247.
• M. J. Tomkinson, Groups covered by finitely many cosets or subgroups, Comm. Algebra, 15(4) (1987), 845–859.
• M. Zarrin, On element centralizers in finite groups, Arch. Math., 93 (2009), –503.
• M. Zarrin, Criteria for the solubility of finite groups by its centralizers, Arch. Math., 96 (2011), 225–226.
• M. Zarrin, On solubility of groups with finitely many centralizers , Bull. Iranian Math. Soc., to appear. Sekhar Jyoti Baishya Department of Mathematics North-Eastern Hill University Permanent Campus, Shillong-793022 Meghalaya, India e-mail: [e-mail]