Archiv der Mathematik

, Volume 102, Issue 4, pp 301–312 | Cite as

Finite simple 3′-groups are cyclic or Suzuki groups

Article

Abstract

In this note we prove that all finite simple 3′-groups are cyclic of prime order or Suzuki groups. This is well known in the sense that it is mentioned frequently in the literature, often referring to unpublished work of Thompson. Recently an explicit proof was given by Aschbacher [3], as a corollary of the classification of \({\mathcal{S}_3}\) -free fusion systems. We argue differently, following Glauberman’s comment in the preface to the second printing of his booklet [8]. We use a result by Stellmacher (see [12]), and instead of quoting Goldschmidt’s result in its full strength, we give explicit arguments along his ideas in [10] for our special case of 3′-groups.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alperin J.L., Gorenstein D.: The multiplicators of certain simple groups. Proc. Am. Math. Soc. 17, 515–519 (1966)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Aschbacher M.: A condition for the existence of a strongly embedded subgroup. Proc. Am. Math. Soc. 38, 509–511 (1973)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    M. Aschbacher, \({\mathcal{S}_3}\) -free 2-fusion systems, Proc. Edinb. Math. Soc., series 2 56 (2013), 27–48.Google Scholar
  4. 4.
    Bender H.: Transitive Gruppen gerader Ordnung, in denen jede Involution genau einen Punkt festläßt. J. Algebra 17, 527–554 (1971)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Feit W., Thompson J.G.: Solvability of groups of odd order. Pacific J. Math. 13, 775–1029 (1963)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    G. Glauberman, Global and local properties of finite groups, Finite simple groups 1–64. Academic Press, London, 1971.Google Scholar
  7. 7.
    Glauberman G.: A sufficient condition for p-stability. Proc. London Math. Soc. 25, 253–287 (1972)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    G. Glauberman, Factorizations in Local Subgroups of Finite Groups, Regional Conference Series in Mathematics, no. 33. American Mathematical Society, Providence, RI, 1977.Google Scholar
  9. 9.
    Goldschmidt D.M.: Strongly closed 2-subgroups of finite groups. Ann. Math. 102, 475–489 (1975)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Goldschmidt D.M.: 2-fusion in finite groups. Ann. Math. 99, 70–117 (1974)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    D. Gorenstein, R. Lyons, and R. Solomon, The classification of the finite simple groups. Number 2, Mathematical Surveys and Monographs, 40.2. American Mathematical Society, Providence, RI, 1996.Google Scholar
  12. 12.
    Stellmacher B.: A characteristic subgroup of Σ4-free groups. Israel J. Math. 94, 367–379 (1996)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Suzuki M.: On a class of doubly transitive groups. Ann. Math. 75, 105–145 (1962)CrossRefMATHGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Martin-Luther-Universität Halle-Wittenberg Institut für MathematikHalleGermany

Personalised recommendations