Geometric Characterizations in Finite Group Theory

  • E. E. Shult
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 372)

Abstract

Over the past several years one may be able to observe an increasing trend to use purely geometric arguments in the proofs of theorems in the theory of finite groups. The idea is a simple one. In the course of proving a theorem about finite groups, one displays some geometric configuration built out of a finite group G. He then proceeds to characterize the known configuration as being some very familiar geometric object. Because of this, the group G is a subgroup of the group of automorphisms of the geometric object, and this can frequently be used to characterize the group G. A good illustration of this principle would be the Suzuki-O’Nan characterization of the three dimensional projective unitary groups over a finite field by the centralizer of an involution.

Keywords

Zine 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Michael Aschbacher, “Doubly transitive groups in which the stabilizer of two points is abelian”, J. Algebra 18 (1971), 114–136. MR43#2059.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    M. Aschbacher, “On doubly transitive groups of degree n E 2 mod 4 “, Illinois J. Math. 16 (1972), 276–279. MR45#8713.MathSciNetMATHGoogle Scholar
  3. [3]
    M. Aschbacher, “2-transitive groups whose 2-point stabilizer has 2-rank 2 “, submitted.Google Scholar
  4. [4]
    Helmut Bender, “Endliche zweifach transitive Permutationsgruppen, deren Involutionen keine Fixpunkte haben”, Math. Z. 104 (1968), 175–204. MR37#2846MathSciNetCrossRefGoogle Scholar
  5. [5]
    Helmut Bender, “Transitive Gruppen gerader Ordnung, in denen jede Involutionen genau einen Punkt festlasst”, J. Algebra 17 (1971), 527–554. MR44#5370.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    F. Beukenhout and E. Shult, “On the foundations of polar geometry”, submitted.Google Scholar
  7. [7]
    George Glauberman, “Central elements in core-free groups”, J. Algebra 4 (1966), 403–420. MR34#2681.MathSciNetCrossRefGoogle Scholar
  8. [8]
    M. Hale and E. Shult, “Equiangular lines, the graph extension theorem, and transfer in triply transitive groups”, submitted.Google Scholar
  9. [9]
    Marshall Hall, “Automorphisms of Steiner Triple systems”, IBM J. Pes. Develop. 4 (1960), 460–472. MR23#Al282.CrossRefMATHGoogle Scholar
  10. [10]
    Christoph Hering, “Zweifach transitive Permutationsgruppen, in denen zwei die maximale Anzahl von Fixpunkten von Involutionen ist”, Math. Z. 104 (1968), 150–174. MR37#295.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    Christoph Hering, “On subgroups with trivial normalizer intersection”, J. Algebra 20 (1972), 622–629. 2b1.239.20026.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    Christoph Hering, and William M. Kantor, and Gary M. Seitz, “Finite groups with a split BN-pair of rank I”, J. Algebra 20 (1972), 435–475. 7,61.244.20003MathSciNetCrossRefGoogle Scholar
  13. [13]
    William M. Kantor, Michael E. O’Nan and Gary M. Seitz, “2-transitive groups in which the stabilizer of two points is cyclic”, J. Algebra 21 (1972), 17–50.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    P.W.H. Lemmens and J.J. Seidel, “Equiangular lines”, J. Algebra (to appear).Google Scholar
  15. [15]
    M. O’Nan, “A characterization of Ln(q) as a permutation group”, Math. Z. 127 (1972), 301–314.MathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    M. O’Nan, “Normal structure of the one-point stabilizer of a doubly transitive permutation group, I”, submitted.Google Scholar
  17. [17]
    Stanley E. Payne, “Affine representations of generalized quadrangles”, J. Algebra 16 (1970), 473–485. MR42#8381.MathSciNetCrossRefGoogle Scholar
  18. [18]
    J.J. Seidel, “Strongly regular graphs”, Recent Progress in Combinatorics, 185–198 (Proc. Third Waterloo Conf. Combinatorics, 1968; Academic Press, New York, London, 1969). MR40#7148.Google Scholar
  19. [19]
    J.J. Seidel, “On two-graphs and Shult’s characterization of symplectic and orthogonal geometries over GF(2) ‘, Tech. Univ. Eindhoven T.H.-Report 73-WSK-02.Google Scholar
  20. [20]
    Ernest E. Shult, “Characterizations of certain classes of graphs”, J. Ccrrbinatorial Theory Ser. B 13 (1972), 142–167. Zó1.227.05110.CrossRefGoogle Scholar
  21. [21]
    Ernest Shult, “On a class of doubly transitive groups”, Illinois J. Math. 16 (1972), 434–445. MR45#5211.MathSciNetMATHGoogle Scholar
  22. [22]
    E.E. Shult, “On doubly transitive groups of even degree”, submitted.Google Scholar
  23. [23]
    E.E. Shult, “Hall triple systems”, University of Florida, mimeographed notes.Google Scholar
  24. [24]
    G. Tallini, “Ruled graphic systems”, Atti. Cony. Geo. Comb. Perugia (1971), 403–411.Google Scholar
  25. [25]
    D. Taylor, “Regular 2-graphs”, submitted.Google Scholar
  26. [26]
    D. Taylor, Personal communication.Google Scholar
  27. [27]
    J.A. Thas, “On 4-gonal configurations”, Geometrica Dedicata (to appear).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1974

Authors and Affiliations

  • E. E. Shult
    • 1
  1. 1.University of FloridaGainesvilleUSA

Personalised recommendations