Advertisement

Computing with Finite Simple Groups

  • John McKay
Part of the Lecture Notes in Mathematics book series (LNM, volume 372)

Abstract

Leech [9] and Birkhoff and Hall [1] are standard references to computational group theory. The lesser-known Petrick [13] contains many articles on symbolic manipulation and group-theoretic work including a description by Sims of techniques he has developed to compute with very large degree permutation groups. These ideas have been used by him [14] most impressively to prove the existence and uniqueness of Lyons’ simple group of order 51 765 179 004 000 000 = 2837567.11.31.37.67 by constructing a permutation representation of it on the coasts of a subgroup G 2(5) of index 8 835 156. It should be added that Lyons’ group has no proper subgroup larger than G 2(5).

Keywords

Simple Group Maximal Subgroup Algebraic Manipulation Proper Subgroup Finite Simple Group 
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.

References

  1. [1]
    Garrett Birkhoff and Marshall Hall, Jr., Computers in algebra and number theory, Proc. Sympos. Appl. Math. Amer. Math. Soc. and the Soc. Indust. Appl. Math., New York City, 1970, 4 (Amer. Math. Soc., Providence, RhodeIsland, 1971). Z61. 236. 00006.Google Scholar
  2. [2]
    Richard Brauer, “Representations of finite groups”, Lectures on Modern. Mathematics [Ed. T.L. Saaty], 1, pp. 133–175 (John Wiley & Sons, New York, London, 1963). MR31#2314.Google Scholar
  3. [3]
    J.H. Conway, “Three lectures on exceptional groups”, Finite simple groups [ed. M.B. Powell and G. Higman], pp. 215–247 (Academic Press, London and New York, 1971). Zb1. 221. 20014.Google Scholar
  4. [4]
    Larry Finkelstein, “The maximal subgroups of Conway’s group C3 and McLaughlin’s group”, J. Algebra 25 (1973), 58–89.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    L. Finkelstein and A. Rudvalis, “Maximal subgroups of the Hall-Janko-Wales group”, J. Algebra 24 (1973), 486–493.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    Dieter Held, “The simple groups related to M24 ”, J. Algebra 13 (1969), 253–296. MR40#2745.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    G. Higman, “Construction of simple groups from character tables”, Finite simple groups [ed. M.B. Powell and G. Higman], 205–214 ( Academic Press, London and New York, 1971 ). Zb1.221.20014.Google Scholar
  8. [8]
    Graham Higman and John McKay, “On Janko’s simple group of order 50, 232, 960 ”, Bull. London Math. Soc. 1 (1969), 89–94; Correction: Bull. London Math. Soc. 1 (1969), 219. MR40#224.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    John Leech (editor), Computational problems in abstract algebra (Oxford, 1967), (Pergamon Press, Oxford, 1970). MR4045374.Google Scholar
  10. [10]
    Richard Lyons, “Evidence for a new finite simple group”, J. Algebra 20 (1972), 540–569. MR45#8722.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    John McKay, “Subgroups and permutation characters”, Computers in algebra and number theory [ed. Garrett Birkhoff and Marshall Hall, Sr.], Proc. Sympos. Appl. Math. Amer. Math. Soc. and the Soc. Indust. Appl. Math., New York City, 1970, 4, pp. 177–181 ( Amer. Math. Soc., Providence, Rhode Island, 1971 ).Google Scholar
  12. [12]
    Spyros S. Magliveras, “The subgroup structure of the Higman-Sims simple group”, Bull. Amer. Math. Soc. 77 (1971), 535–539. MR44#310.MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    Stanley Roy Petrick (editor), Second Symposium on symbolic and algebraic manipulation,Special Interest Group on Symbolic and Algebraic Manipulation Proceedings (Assoc. for Computing Machinery, 1971).Google Scholar
  14. [14]
    C.C. Sims, to appear.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1974

Authors and Affiliations

  • John McKay
    • 1
  1. 1.School of Computer ScienceMcGill UniversityMontrealCanada

Personalised recommendations