Advertisement

Construction of Finite Matrix Groups

  • Robert A. Wilson
Conference paper
Part of the Progress in Mathematics book series (PM, volume 173)

Abstract

We describe various methods of construction of matrix representations of finite groups. The applications are mainly, but not exclusively, to quasisimple or almost simple groups. Some of the techniques can also be generalized to permutation representations.

Keywords

Conjugacy Class Simple Group Maximal Subgroup Standard Basis Seed Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. [1]
    J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson: An ATLAS of Finite Groups, Clarendon Press, Oxford, 1985.Google Scholar
  2. [2]
    G. Cooperman, L. Finkelstein, M. Tselman and B. York: Constructing permutation representations for large matrix groups, in: Proc. ISSAC-94. ACM Press, 1994.Google Scholar
  3. [3]
    H. Gollan: Über die Konstruktion modularer Darstellungen am Beispiel der einfachen Tits-Gruppe 2F4(2)’in Charakteristik 5, Dissertation, Essen, 1988.zbMATHGoogle Scholar
  4. [4]
    D. G. Higman and C. C. Sims: A simple group of order 44, 352, 000, Math. Z. 105, (1968), 110–113.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    D. F. Holt and S. Rees: Testing modules for irreducibility, J. Austral. Math. Soc. Ser. A 57, (1994), 1–16.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    Z. Janko:A new finite simple group with Abelian Sylow 2-subgroups,and its characterization, J. Algebra 3, (1966), 147–186.MathSciNetzbMATHCrossRefGoogle Scholar

Bibliography

  1. [7]
    C. Jansen, K. Lux, R. A. Parker and R. A. Wilson: An Atlas of Brauer Characters, Clarendon Press, Oxford, 1995.zbMATHGoogle Scholar
  2. [8]
    C. Jansen and R. A. Wilson: Two new constructions of the O’Nan group,J. London Math. Soc. 56, (1997), 579–583.MathSciNetzbMATHGoogle Scholar
  3. [9]
    C. Jansen and R. A. Wilson: The 2- and 3-modular characters of the O’Nan group,J. London Math. Soc. 57, (1998), 71–90.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [10]
    C. Jansen and R. A. Wilson: The minimal faithful 3-modular representation for the Lyons group, Comm Algebra 24, (1996), 873–879.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [11]
    D. L. Johnson: Topics in the theory of group presentations, LMS Lecture Note Series 42, Cambridge University Press, 1980.zbMATHCrossRefGoogle Scholar
  6. [12]
    S. A. Linton, R. A. Parker, P. G. Walsh and R. A. Wilson: Computer construction of the Monster, J.Group Theory, to appear.Google Scholar
  7. [13]
    D. Livingstone: On a permutation representation of the Janko group, J. Algebra 6, (1967), 43–55.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [14]
    W. Meyer, W. Neutsch and R. Parker: The minimal 5-representation of Lyons’ sporadic groupMath. Ann. 272, (1985), 29–39.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [15]
    S. P. Norton: The construction of J 4, in: The Santa Cruz conference on finite groups (eds. B. Cooperstein and G. Mason), pp. 271–277. Proceedings of Symposia in Pure Mathematics, Vol. 37, American Mathematical Society,1980.Google Scholar
  10. [16]
    R. A. Parker: The computer calculation of modular characters (The `Meataxe’), in: Computational Group Theory (ed. M. D. Atkinson), Academic Press, 1984, pp. 267–274.Google Scholar
  11. [17]
    R. A. Parker: An integral Meat-axe, in: The Atlas 10 years on (eds. R. T. Curtis and R. A. Wilson), pp. 215–228. LMS Lecture Note Series 249, Cambridge University Press, 1998.CrossRefGoogle Scholar
  12. [18]
    R. A. Parker and R. A. Wilson: Computer construction of matrix representations of finite groups over finite fields, J. Symbolic Comput. 9, (1990), 583–590.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [19]
    M. Ringe: The C Meataxe 2.3,documentation„ RWTH Aachen, 1995.Google Scholar
  14. [20]
    A. Ryba and R. Wilson: Matrix generators for the Harada-Norton group, Experimental Math. 3, (1994), 137–145.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [21]
    M. Schönert et al.: GAP 3.4 Manual (Groups, Algorithms,and Programming), RWTH Aachen, 1994.Google Scholar
  16. [22]
    I. A. Suleiman and R. A. Wilson: Computer construction of matrix representations of the covering group of the Higman—Sims group, J. Algebra 148, (1992), 219–224.MathSciNetzbMATHCrossRefGoogle Scholar

Robert A. Wilson

  1. [23]
    I. A. Suleiman and R. A. Wilson: Construction of the fourfold cover of the Mathieu group M 22,Experimental Math. 2, (1993), 11–14.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [24]
    I. A. Suleiman and R. A. Wilson: Covering and automorphism groups of U 6(2), Quart. J. Math. (Oxford) 48 (1998), 511–517.MathSciNetCrossRefGoogle Scholar
  3. [25]
    I. A. Suleiman and R. A. Wilson: Construction of exceptional covers of generic groups, Math. Proc. Cambridge Philos. Soc., to appear (1998).Google Scholar
  4. [26]
    J. G. Thompson: Finite-dimensional representations of free products with an amalgamated subgroup, J. Algebra 69 (1981), 146–149.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [27]
    P. G. Walsh: Computational study of the Monster and other sporadic groups, Ph.D. thesis, Birmingham, 1996.Google Scholar
  6. [28]
    R. A. Wilson: Standard generators for sporadic simple groups, J. Algebra 184, (1996), 505–515.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [29]
    R. A. Wilson: A new construction of the Baby Monster, and its applications, Bull. London Math. Soc. 25 (1993), 431–437.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [30]
    R. A. Wilson: Some new subgroups of the Baby Monster,Bull. London Math. Soc. 25 (1993), 23–28.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [31]
    R. A. Wilson: More on the maximal subgroups of the Baby Monster,Arch. Math. (Basel) 61 (1993), 497–507.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [32]
    R. A. Wilson: The maximal subgroups of the Baby Monster, I, J. Algebra, to appear.Google Scholar
  11. [33]
    R. A. Wilson: Matrix generators for Fischer’s group Fi 24 Math. Proc. Cambridge Philos. Soc.113 (1993), 5–8.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [34]
    R. A. Wilson: A construction of the Lyons group in GL 2480(4)and a new uniqueness proof, Arch. Math. (Basel) 70, (1998), 11–15.zbMATHCrossRefGoogle Scholar
  13. [35]
    R. A. Wilson et al.: A world-wide-web atlas of group representations,http://www.mat.bham.ac.uk/atlas/

Copyright information

© Springer Basel AG 1999

Authors and Affiliations

  • Robert A. Wilson

There are no affiliations available

Personalised recommendations