Computing with Finite Simple Groups
Leech  and Birkhoff and Hall  are standard references to computational group theory. The lesser-known Petrick  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  most impressively to prove the existence and uniqueness of Lyons’ simple group of order 51 765 179 004 000 000 = 2837518.104.22.168.67 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).
KeywordsSimple Group Maximal Subgroup Algebraic Manipulation Proper Subgroup Finite Simple Group
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- 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
- 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
- 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
- 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
- John Leech (editor), Computational problems in abstract algebra (Oxford, 1967), (Pergamon Press, Oxford, 1970). MR4045374.Google Scholar
- 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
- 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
- C.C. Sims, to appear.Google Scholar