Algebra and Logic

, Volume 52, Issue 1, pp 1–14 | Cite as

Recognizability of groups G 2(q) by spectrum


Two groups are said to be isospectral if they have equal sets of element orders. It is proved that for every finite simple exceptional group L = G 2(q) of Lie type, any finite group G isospectral to L must be isomorphic to L.


finite simple group exceptional group of Lie type element order spectrum of group recognition by spectrum 


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  1. 1.
    V. D. Mazurov, “Groups with prescribed orders of elements,” Izv. Ural. Gos. Univ., Mat. Mekh., No. 36 (Mat. Mekh., iss. 7), 119–138 (2005).Google Scholar
  2. 2.
    M. A. Grechkoseeva, W. J. Shi, and A. V. Vasil’ev, “Recognition by spectrum of finite simple groups of Lie type,” Front. Math. China, 3, No. 2, 275–285 (2008).MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    A. V. Vasil’ev, “Recognizability of groups G 2(3n) by their element orders,” Algebra Logika, 41, No. 2, 130–142 (2002).MathSciNetMATHGoogle Scholar
  4. 4.
    V. D. Mazurov, “Recognition of finite simple groups S 4(q) by their element orders,” Algebra Logika, 41, No. 2, 166–198 (2002).MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    A. V. Zavarnitsine, “Recognition of finite groups by the prime graph,” Algebra Logika, 45, No. 4, 390–408 (2006).MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    V. D. Mazurov, “Recognition of finite groups by a set of orders of their elements,” Algebra Logika, 37, No. 6, 651–666 (1998).MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    K. Zsigmondi, “Zur Theorie der Potenzreste,” Mon. Math. Phys., 3, 265–284 (1892).CrossRefGoogle Scholar
  8. 8.
    M. Roitman, “On Zsigmondy primes,” Proc. Am. Math. Soc., 125, No. 7, 1913–1919 (1997).MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of Finite Groups, Clarendon Press, Oxford (1985).MATHGoogle Scholar
  10. 10.
    J. S. Williams, “Prime graph components of finite groups,” J. Alg., 69, No. 2, 487–513 (1981).MATHCrossRefGoogle Scholar
  11. 11.
    A. V. Vasil’ev, “On connection between the structure of a finite group and the properties of its prime graph,” Sib. Mat. Zh., 46, No. 3, 511–522 (2005).MATHGoogle Scholar
  12. 12.
    A. V. Vasil’ev and I. B. Gorshkov, “On recognition of finite simple groups with connected prime graph,” Sib. Mat. Zh., 50, No. 2, 292–299 (2009).MathSciNetMATHGoogle Scholar
  13. 13.
    A. S. Kondratiev, “Prime graph components of finite simple groups,” Mat. Sb., 180, No. 6, 787–797 (1989).Google Scholar
  14. 14.
    A. V. Vasiliev and E. P. Vdovin, “An adjacency criterion for the prime graph of a finite simple group,” Algebra Logika, 44, No. 6, 682–725 (2005).MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    W. M. Kantor and A. Seress, “Prime power graphs for groups of Lie type,” J. Alg., 247, 370–434 (2002).MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    D. Deriziotis, Conjugacy Classes and Centralizers of Semisimple Elements in Finite Groups of Lie Type, Vorlesungen Fachbereich Math. Univ. Essen, 11 (1984).Google Scholar
  17. 17.
    A. A. Buturlakin and M. A. Grechkoseeva, “The cyclic structure of maximal tori of the finite classical groups,” Algebra Logika, 46, No. 2, 129–156 (2007).MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    A. M. Staroletov, “Sporadic composition factors of finite groups isospectral to simple groups,” Sib. El. Mat. Izv., 8, 268–272 (2011).MathSciNetGoogle Scholar
  19. 19.
    A. V. Zavarnitsine, “Recognition of the simple groups L 3(q) by element orders,” J. Group Th., 7, No. 1, 81–97 (2004).MathSciNetMATHGoogle Scholar
  20. 20.
    A. V. Vasil’ev, M. A. Grechkoseeva, V. D. Mazurov, Kh. P. Chao, G. Yu. Chen, and W. Shi, “Recognition of the finite simple groups F 4(2m) by spectrum,” Sib. Mat. Zh., 45, No. 6, 1256–1262 (2004).MathSciNetMATHGoogle Scholar
  21. 21.
    V. D. Mazurov, “Characterizations of finite groups by sets of orders of their elements,” Algebra Logika, 36, No. 1, 37–53 (1997).MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    R. M. Guralnick and P. H. Tiep, “Finite simple unisingular groups of Lie type,” J. Group Theory, 6, No. 3, 271–310 (2003).MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    R. W. Carter, Simple Groups of Lie Type, Pure Appl. Math., 28, Wiley, London (1972).MATHGoogle Scholar
  24. 24.
    I. D. Suprunenko and A. E. Zalesski, “Fixed vectors for elements in modules for algebraic groups,” Int. J. Alg. Comp., 17, Nos. 5/6, 1249–1261 (2007).MathSciNetMATHCrossRefGoogle Scholar

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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia

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