Advertisement

Mathematical Notes

, Volume 104, Issue 5–6, pp 696–701 | Cite as

Finite Groups without Elements of Order Six

  • A. S. Kondrat’evEmail author
  • N. A. Minigulov
Article
  • 8 Downloads

Abstract

In 1977, in three papers by Podufalov, by Gordon, and by Fletcher, Stellmacher, and Stewart, finite simple groups without elements of order 6 were determined independently. In the present paper, using this result, we obtain a sufficiently complete description of the structure of a general finite group with this property.

Keywords

finite group element of order 6 prime graph 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    N. D. Podufalov, “Finite simple groups without elements of order six,” Algebra Logika 16 (2), 200–203 (1977) [Algebra Logic 16 (2), 133–135 (1978)].MathSciNetCrossRefGoogle Scholar
  2. 2.
    L. F. Fletcher, B. Stellmacher, and W. B. Stewart, “Endliche Gruppen, die kein Element der Ordnung 6 enthalten,” Quart. J.Math. Oxford Ser. (2) 28 (110), 143–154 (1977).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    L. M. Gordon, “Finite simple groups with no elements of order six,” Bull. Austral.Math. Soc. 17 (2), 235–246 (1977).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    M. Suzuki, “On a class of doubly transitive groups,” Ann. Math. 75 (1), 105–145 (1962).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    G. Higman, Odd Characterizations of Finite Simple Groups, Lecture Notes (University of Michigan, Michigan, 1968).Google Scholar
  6. 6.
    W. B. Stewart, “Groups having strongly self-centralizing 3-centralizers,” Proc. London Math. Soc. (3) 26, 653–680 (1973).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    A. S. Kondrat’ev, “Finite groups with given properties of their prime graphs,” Algebra Logika 55 (1), 113–120 (2016) [Algebra Logic 55 (1), 77–82 (2016)].MathSciNetGoogle Scholar
  8. 8.
    D. Gorenstein, R. Lyons, and R. Solomon, The Classification of the Finite Simple Groups, Number 3 (Amer.Math. Soc., Providence, RI, 1994).CrossRefzbMATHGoogle Scholar
  9. 9.
    J. N. Bray, D. F. Holt, and C. M. Roney-Dougal, The Maximal Subgroups of the Low-Dimensional Finite Classical Groups (Cambridge Univ. Press, Cambridge, 2013).CrossRefzbMATHGoogle Scholar
  10. 10.
    M. Aschbacher, Finite Group Theory (Cambridge Univ. Press, Cambridge, 1986).zbMATHGoogle Scholar
  11. 11.
    J. N. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of Finite Groups (Clarendon Press, Oxford, 1985).zbMATHGoogle Scholar
  12. 12.
    D. Gorenstein, Finite Groups (Harper and Row, New York, 1968).zbMATHGoogle Scholar
  13. 13.
    V. D. Mazurov, “Characterizations of finite groups by sets of orders of their elements,” Algebra Logika 36 (1), 37–53 (1997) [Algebra Logic 36 (1), 23–32 (1997)].MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    B. Hartley and T. Meixner, “Finite soluble groups containing an element of prime order whose centralizer is small,” Arch. Math. (Basel) 36 (3), 211–213 (1981).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    G. Higman, “Finite groups in which every element has prime power order,” J. London Math. Soc. (2) 32, 335–342 (1957).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Institute of Mathematics and MechanicsUral Branch of the Russian Academy of SciencesYekaterinburgRussia
  2. 2.Yeltsyn Ural Federal UniversityYekaterinburgRussia

Personalised recommendations