ON FINITE GROUPS WITH SPECIFIC NUMBER OF CENTRALIZERS

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

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.
2. A. R. Ashrafi, On finite groups with a given number of centralizers, Algebra Colloq., 7(2) (2000), 139–146.
3. A. R. Ashrafi, Counting the centralizers of some finite groups, Korean J. Com- put. Appl. Math., 7(1) (2000), 115–124.
4. 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.
5. 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.
6. S. M. Belcastro and G. J. Sherman, Counting centralizers in finite groups, Math. Magazine, 67(5) (1994), 366–374.
7. Y. G. Berkovich and E. M. Zhmu´d, Characters of Finite Groups, Part 1, Transl. Math. Monographs 172, Amer. Math. Soc., Providence. RI, 1998.
8. A. Castelaz, Commutativity degree of finite groups, MSc. Thesis, Wake Forest University, North Carolina, May, 2010.

9. J. H. E. Cohn, On n-sum groups, Math. Scand., 75 (1994), 44–58.
10. J. D. Dixon, Problems in Group Theory, Dover Publications, Inc., Mineola, New York, 2007.
11. 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–
12. H. Kurzweil and B. Stellmacher, The Theory of Finite Groups, An Introduc- tion, Springer-Verlag, New York, Inc. 2004.
13. D. J. Rusin, What is the probability that two elements of a finite group com- mute?, Pacific J. Math., 82(1) (1979), 237–247.
14. M. J. Tomkinson, Groups covered by finitely many cosets or subgroups, Comm. Algebra, 15(4) (1987), 845–859.
15. M. Zarrin, On element centralizers in finite groups, Arch. Math., 93 (2009), –503.
16. M. Zarrin, Criteria for the solubility of finite groups by its centralizers, Arch. Math., 96 (2011), 225–226.
17. 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]