Algebra and Logic

, Volume 44, Issue 6, pp 381–406 | Cite as

An Adjacency Criterion for the Prime Graph of a Finite Simple Group

  • A. V. Vasiliev
  • E. P. Vdovin
Article

Abstract

For every finite non-Abelian simple group, we give an exhaustive arithmetic criterion for adjacency of vertices in a prime graph of the group. For the prime graph of every finite simple group, this criterion is used to determine an independent set with a maximal number of vertices and an independent set with a maximal number of vertices containing 2, and to define orders on these sets; the information obtained is collected in tables. We consider several applications of these results to various problems in finite group theory, in particular, to the recognition-by-spectra problem for finite groups.

Keywords

finite group finite simple group group of Lie type spectrum of a finite group recognition by spectrum prime graph of a finite group independence number of a prime graph 2-independence number of a prime graph 

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REFERENCES

  1. 1.
    J. S. Williams, “Prime graph components of finite groups,” J. Alg., 69, No.2, 487–513 (1981).CrossRefMATHGoogle Scholar
  2. 2.
    A. S. Kondratiev, “On prime graph components for finite simple groups,” Mat. Sb., 180, No.6, 787–797 (1989).Google Scholar
  3. 3.
    V. D. Mazurov, “Characterization of groups by arithmetic properties,” Alg. Coll., 11, No.1, 129–140 (2004).MATHMathSciNetGoogle Scholar
  4. 4.
    A. V. Vasiliev, “On connection between the structure of a finite group and properties of its prime graph,” Sib. Mat. Zh., 46, No.3, 511–522 (2005).MathSciNetGoogle Scholar
  5. 5.
    J. Conway, R. Curtis, S. Norton, et al., Atlas of Finite Groups, Clarendon, Oxford (1985).MATHGoogle Scholar
  6. 6.
    The GAP Group, GAP — Groups, Algorithms, and Programming, Vers. 4.4 (2004); http://www.gap-system.org.Google Scholar
  7. 7.
    R. W. Carter, Simple Groups of Lie Type, Pure Appl. Math., 28, Wiley, London (1972).Google Scholar
  8. 8.
    J. E. Humphreys, Linear Algebraic Groups, Springer, New York (1972).Google Scholar
  9. 9.
    R. Steinberg, Endomorphisms of Algebraic Groups, Mem. Am. Math. Soc., Vol. 80 (1968).Google Scholar
  10. 10.
    A. Borel and J. de Siebental, “Les-sous-groupes fermes de rang maximum des groupes de Lie clos,” Comm. Math. Helv., 23, 200–221 (1949).MATHGoogle Scholar
  11. 11.
    E. B. Dynkin, “Semisimple subalgebras of semisimple Lie algebras,” Mat. Sb., 30, No.2, 349–462 (1952).MATHMathSciNetGoogle Scholar
  12. 12.
    R. W. Carter, Finite Groups of Lie Type: Conjugacy Classes and Complex Characters, Wiley, New York (1985).MATHGoogle Scholar
  13. 13.
    R. Carter, “Centralizers of semisimple elements in finite classical groups,” Proc. London Math. Soc., III. Ser., 42, No.1, 1–41 (1981).MATHMathSciNetGoogle Scholar
  14. 14.
    R. W. Carter, “Conjugacy classes in the Weyl group,” Comp. Math., 25, No.1, 1–59 (1972).MATHMathSciNetGoogle Scholar
  15. 15.
    D. Deriziotis, “Conjugacy classes and centralizers of semisimple elements in finite groups of Lie type,” Vorlesungen aus dem Fachbereich Mathetmatic der Universitat Essen, Heft 11 (1984).Google Scholar
  16. 16.
    K. Zsigmondi, “Zur Theorie der Potenzreste,” Mon. Math. Phys., 3, 265–284 (1892).Google Scholar
  17. 17.
    D. I. Deriziotis, “The centralizers of semisimple elements of the Chevalley groups E 7 and E 8,” Tokyo J. Math., 6, No.1, 191–216 (1983).MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    M. S. Lucido and A. R. Moghaddamfar, “Groups with complete prime graph connected components,” J. Group Theory, 7, No.3, 373–384 (2004).MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • A. V. Vasiliev
  • E. P. Vdovin

There are no affiliations available

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