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On F-groups with the central factor of order p4

  • Seyyed Majid Jafarian Amiri EMAIL logo , Halimeh Madadi und Hojjat Rostami
Veröffentlicht/Copyright: 22. September 2017
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Mathematica Slovaca
Aus der Zeitschrift Mathematica Slovaca Band 67 Heft 5

Abstract

A finite group G is called an F-group (G ∈ F) if for every x, y ∈ G ∖ Z(G), CG(x) ≤ CG(y) implies that CG(x) = CG(y). An important subclass of F-groups are CA-groups, consisting of groups in which all centralizers of noncentral elements are abelian. In this paper, among other results, we find the number of element centralizers and the maximum cardinality of subsets of pairwise non-commuting elements in an F-group G with |GZ(G)| = p4 for some prime p.

MSC 2010: 20D15
Keywords: p-group; F-group; non-commuting element; centralizer

(Communicated by Vincenzo Marra)

Acknowledgement

The authors thank the anonymous referee for his/her useful comments and suggestions which substantially improved the paper.

References

[1] Abdollahi, A.—Jafarian Amiri, S. M.—Mohammadi Hassanabadi, A.: Groups with specific number of centralizers, Houston J. Math. 33 (2007), 43–57.Suche in Google Scholar

[2] Abdollahi, A.—Akbari, S.—Maimani, H. R.: Non-commuting graph of a group, J. Algebra 298 (2006), 468–492.10.1016/j.jalgebra.2006.02.015Suche in Google Scholar

[3] Ashrafi, A. R.: On finite groups with a given number of centralizers, Algebra Colloq. 7 (2000), 139–146.10.1007/s10011-000-0139-5Suche in Google Scholar

[4] Azad, A.—Iranmanesh, M. A.—Praeger, C. E.—Spiga, P.: Abelian coverings of finite general linear groups and an application to their non-commuting graphs, J. Algebraic Comb. 34 (2011), 683–710.10.1007/s10801-011-0288-2Suche in Google Scholar

[5] Azad, A.—Praeger, C. E.: Maximal subsets of pairwise non-commuting elements of three-dimensional general linear groups, Bull. Aus. Math. Soc. 80 (2009), 91–104.10.1017/S0004972709000057Suche in Google Scholar

[6] Azad, A.—Fouladi, S.— Orfi, R.: Maximal subsets of pairwise non-commuting elements of some finite p- groups, Bull. Iran Math. Soc. 39 (2013), 187–192.Suche in Google Scholar

[7] Baer, R.: Groups with preassigned central and central quotient group, Trans. Amer. Math. Soc. 44 (1938), 387–412.10.1090/S0002-9947-1938-1501973-3Suche in Google Scholar

[8] Baishya, S. J.: On finite groups with specific number of centralizers, Int. Electron. J. Algebra 13 (2013), 53–62.Suche in Google Scholar

[9] Belcastro, S. M.—Sherman, G. J.: Counting centralizers in finite groups, Math. Magazine 67 (1994), 366–374.10.2307/2690998Suche in Google Scholar

[10] Berkovich, Y. G.—Zhmud, E. M.: Characters of Finite Groups, I. Trans. Math. Monographs 172, Amer. Math. Soc., Providence. RI, 1998.10.1090/mmono/181Suche in Google Scholar

[11] Berkovich, Y. G.: Groups of Prime Power Order, I, Walter de Gruyter, 2008.10.1515/9783110208238Suche in Google Scholar

[12] Bertram, E. A.: Some applications of graph theory to finite groups, Discrete Math. 44 (1983), 31–43.10.1016/0012-365X(83)90004-3Suche in Google Scholar

[13] Blackburn, N.: Groups of prime-power order having an abelian centralizer of type (r, 1), Monatsh. Math. 99 (1985), 1–18.10.1007/BF01300734Suche in Google Scholar

[14] Chin, A. Y. M.: On non-commuting sets in an extraspecial p-group, J. Group Theory 8 (2005), 189–194.10.1515/jgth.2005.8.2.189Suche in Google Scholar

[15] Dixon, J. D.: Problems in Group Theory, Blaisdell, Waltham, MA, 1967.Suche in Google Scholar

[16] Dolfi, S.—Herzog, M.—Jabara, E.: Finite groups whose noncentral commuting elements have centralizers of equal size, Bull. Aust. Math. Soc. 82 (2010), 293–304.10.1017/S0004972710000298Suche in Google Scholar

[17] Fouladi, S.—Orfi, R.: Maximum size of subsets of pairwise noncommuting elements in finite metacyclic p-groups, Bull. Aust. Math. Soc. 87 (2013), 18–23.10.1017/S0004972712000111Suche in Google Scholar

[18] Fouladi, S.—Orfi, R.: Maximal subsets of pairwise noncommuting elements of some p-groups of maximal class, Bull. Aust. Math. Soc. 84 (2011), 447–451.10.1017/S0004972711002401Suche in Google Scholar

[19] Isaacs, I. M.: Finite Group Theory. Grad. Stud. Math, Providence, R. I., 2008.10.1090/gsm/092Suche in Google Scholar

[20] Isaacs, I. M.: Subgroups generated by small classes in finite groups, Proc. Amer. Math. Soc. 136 (2008), 2299–2301.10.1090/S0002-9939-08-09263-0Suche in Google Scholar

[21] Ito, N.: On finite groups with given conjugate type, I, Nagoya J. Math. 6 (1953), 17–28.10.1017/S0027763000016937Suche in Google Scholar

[22] Jafarian Amiri, S. M.—Madadi, H.—Rostami, H.: On 9-centraliser groups, J. Algebra Appl. 14 (2015), 1–13.10.1142/S0219498815500036Suche in Google Scholar

[23] Neumann, B. H.: A problem of Paul Erdös on groups, J. Austral. Math. Soc. Ser. A 21 (1976), 467–472.10.1017/S1446788700019303Suche in Google Scholar

[24] Pyber, L.: The number of pairwise non-commuting elements and the index of the center in a finite group, J. London Math. Soc. 35 (1987), 287–295.10.1112/jlms/s2-35.2.287Suche in Google Scholar

[25] Rebmann, J.: F-Gruppen, Arch. Math. 22 (1971), 225–230.10.1007/BF01222567Suche in Google Scholar

[26] Rocke, D. M.: P-groups with abelian centralizers, Proc. London Math. Soc. 30 (1975), 55–57.10.1112/plms/s3-30.1.55Suche in Google Scholar

[27] Schmidt, R.: Zentralisatorverbande endlicher Gruppen, Rend. Sem. Mat. Univ. Padova 44 (1970), 97–131.Suche in Google Scholar

[28] Shahriari, S.: On normal subgroups of capable groups, Arch. Math. 48 (1987), 193–198.10.1007/BF01195351Suche in Google Scholar

[29] Zarrin, M.: On element-centralizers in finite groups, Arch. Math. 93 (2009), 497–503.10.1007/s00013-009-0060-1Suche in Google Scholar

[30] Zarrin, M.: Criteria for the solubility of finite groups by its centralizers, Arch. Math. 96 (2011), 225–226.10.1007/s00013-011-0233-6Suche in Google Scholar

Received: 2015-11-19
Accepted: 2015-12-1
Published Online: 2017-9-22
Published in Print: 2017-10-26

© 2017 Mathematical Institute Slovak Academy of Sciences

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https://doi.org/10.1515/ms-2017-0038
Schlagwörter für diesen Artikel
p-group; F-group; non-commuting element; centralizer

Artikel in diesem Heft

  2. Cyclic and rotational latin hybrid triple systems
  4. Notes on mildly distributive semilattices
  6. On generalized completely distributive posets
  8. Properties of non-associative MV-algebras
  10. On the upper and lower exponential density functions
  12. Quadratic permutations, complete mappings and mutually orthogonal latin squares
  14. On F-groups with the central factor of order p4
  16. Comparison of some families of real functions in porosity terms
  18. Negative interest rates: why and how?
  20. Homoclinic and heteroclinic motions in hybrid systems with impacts
  22. Some fixed point theorems in Branciari metric spaces
  24. S-essential spectra and measure of noncompactness
  26. Unified approach to graphs and metric spaces
  28. On structural properties of porouscontinuous functions
  30. A class of topological spaces between the classes of regular and urysohn spaces
  32. On The betti numbers of oriented Grassmannians and independent semi-invariants of binary forms
  34. Generalized Baskakov type operators
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