Journal of Applied Mathematics and Computing

, Volume 22, Issue 1–2, pp 403–410 | Cite as

On finite groups with exactly seven element centralizers



For a finite groupG, #Cent(G) denotes the number of centralizers of its elements. A groupG is calledn-centralizer if #Cent(G) =n, and primitive n-centralizer if\(\# Cent(G) = \# Cent\left( {\frac{G}{{Z(G)}}} \right) = n\).

The first author in [1], characterized the primitive 6-centralizer finite groups. In this paper we continue this problem and characterize the primitive 7-centralizer finite groups. We prove that a finite groupG is primitive 7-centralizer if and only if\(\frac{G}{{Z(G)}} \cong D_{10} \) orR, whereR is the semidirect product of a cyclic group of order 5 by a cyclic group of order 4 acting faithfully. Also, we compute#Cent(G) for some finite groups, using the structure ofG modulu its center.

AMS Mathematics Subject Classification

20D99 20E07 

Key words and phrases

Finite group n-centralizer group primitiven-centralizer group 


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  1. 1.
    A. R. Ashrafi,On finite groups with a given number of centralizers, Algebra Coloquium,7(2) (2000), 139–146.MATHMathSciNetGoogle Scholar
  2. 2.
    A. R. Ashrafi,Counting the centralizers of some finite groups, J. Appl. Math. & Computing(old:KJCAM)7(1) (2000), 115–124.MATHMathSciNetGoogle Scholar
  3. 3.
    S. M. Belcastro and G. J. Sherman,Counting Centralizers in Finite Groups, Math. Mag.5 (1994), 366–374.MathSciNetCrossRefGoogle Scholar
  4. 4.
    R. A. Bryce, V. Fedri and L. Serena,Covering groups with subgroups, Bull. Austral. Math. Soc.55(3) (1997), 469–476.MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    J. H. E. Cohn,On n-sum groups, Math. Scand.75 (1994), 44–58.MATHMathSciNetGoogle Scholar
  6. 6.
    M. R. Darafsheh and Z. Mostaghim,Computation of the complex characters of the group AUT(GL 7(2)), J. Appl. Math. & Computing(old:KJCAM)4 (1997), 193–210.MATHMathSciNetGoogle Scholar
  7. 7.
    M. R. Darafsheh and F. Nowroozi Larki,The character table of the group GL 2(q) when extended by a certain group of order two, J. Appl. Math. & Computing(old:KJCAM)7 (2000), 643–654.MATHMathSciNetGoogle Scholar
  8. 8.
    S. Haber and A. Rosenfeld,Groups as unions of proper subgroups, Amer. Math. Monthly66 (1959), 491–494.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    N. Ito,On the degrees of irreducible representations of a finite group, Nagoya Math. J.3 (1951) 5–6.MATHMathSciNetGoogle Scholar
  10. 10.
    G. Karpilovsky,Group Representations, Volume 2, North-Holland Mathematical Studies,177, Amesterdam-New York-Oxford-Tokyo.Google Scholar
  11. 11.
    B. H. Neumann,Groups covered by permutable subsets, J. London Math. Soc.29 (1954), 236–248.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    D. J. S. Robinson,A course in the theory of groups, Springer-Verlag New York, 1980.Google Scholar
  13. 13.
    M. Schonert et al.,GAP: Groups, algorithms, and programming, Lehrstuhl D fur Mathematik, RWTH Aachen, 1994.Google Scholar
  14. 14.
    M. J. Tomkinson,Groups covered by finitely many cosets or subgroups, Comm. Alg.15 (1987), 845–859.MATHMathSciNetGoogle Scholar

Copyright information

© Korean Society for Computational & Applied Mathematics and Korean SIGCAM 2006

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceUniversity of KashanKashanIran
  2. 2.Department of MathematicsIsfahan University of TechnologyIsfahanIran

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