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Siberian Mathematical Journal

, Volume 46, Issue 2, pp 325–332 | Cite as

On the noncommuting graph associated with a finite group

  • A. R. Moghaddamfar
  • W. J. Shi
  • W. Zhou
  • A. R. Zokayi
Article

Abstract

Let G be a finite group. We define the noncommuting graph ∇(G) as follows: the vertex set of ∇(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. We study some properties of ∇(G) and prove that, for many groups G, if H is a group with ∇(G) isomorphic to ∇(H) then |G| = |H|.

Keywords

group noncommuting graph regular graph 

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References

  1. 1.
    Williams J. S., “Prime graph components of finite groups,” J. Algebra, 69, 487–513 (1981).Google Scholar
  2. 2.
    Abe S. and Iiyori N., “A generalization of prime graphs of finite groups,” Hokkaido Math. J., 29, No.2, 391–407 (2000).Google Scholar
  3. 3.
    Segev Y., “On finite homomorphic images of the multiplicative group of a division algebra,” Ann. of Math., 149, 219–251 (1999).Google Scholar
  4. 4.
    Segev Y. and Seitz G., “Anisotropic groups of type A n and the commuting graph of finite simple groups,” Pacific J. Math., 202, 125–225 (2002).Google Scholar
  5. 5.
    Kondrat’ev A. S., “On prime graph components of finite simple groups,” Mat. Sb., 180, No.6, 787–797 (1989).Google Scholar
  6. 6.
    Kondrat’ev A. S. and Mazurov V. D., “Recognition of alternating groups of prime degree from their element orders,” Siberian Mat. J., 41, No.2, 294–302 (2000).Google Scholar
  7. 7.
    Mazurov V. D., “Recognition of finite simple groups S 4(q) by their element orders,” Algebra and Logic, 41, No.2, 93–110 (2002).Google Scholar
  8. 8.
    Ito N., “On finite groups with given conjugate types I,” Nagoya Math. J., 6, 17–28 (1953).Google Scholar
  9. 9.
    Conway J. H., Curtis R. T., Norton S. P., Parker R. A., and Wilson R. A., Atlas of Finite Groups, Clarendon Press, Oxford (1985).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • A. R. Moghaddamfar
    • 1
  • W. J. Shi
    • 2
    • 3
  • W. Zhou
    • 2
    • 3
  • A. R. Zokayi
    • 1
  1. 1.K. N. Toosi University of TechnologyTehranIran
  2. 2.Southwest Normal UniversityChongqingPeople’s Republic of China
  3. 3.Soochow UniversitySuzhouPeople’s Republic of China

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