On an algorithm for finding a base and a strong generating set for a group given by generating permutations

Author:
Jeffrey S. Leon

Journal:
Math. Comp. **35** (1980), 941-974

MSC:
Primary 20-04; Secondary 20F05

DOI:
https://doi.org/10.1090/S0025-5718-1980-0572868-5

MathSciNet review:
572868

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Abstract: This paper deals with the problem of finding a base and strong generating set for the group generated by a given set of permutations. The concepts of base and strong generating set were introduced by Sims [5], [6] and provide the most effective tool for computing with permutation groups of high degree. One algorithm, originally proposed by Sims [7], is described in detail; its behavior on a number of groups is studied, and the influence of certain parameters on its performance is investigated. Another algorithm, developed by the author, is given, and it is shown how the two algorithms may be combined to yield an exceptionally fast and effective method.

**[1]**JOHN J. CANNON, LUCIEN A. DIMINO, GEORGE HAVAS & JANE M. WATSON, "Implementation and analysis of the Todd-Coxeter algorithm,"*Math. Comp.*, v. 27, 1973, pp. 463-490. MR**0335610 (49:390)****[2]**JOHN J. CANNON & GEORGE HAVAS, "Defining relations for the Held-Higman-Thompson simple group,"*Bull. Austral. Math. Soc.*, v. 11, 1974, pp. 43-46. MR**0360795 (50:13242)****[3]**MARSHALL HALL, JR.,*The Theory of Groups*, Macmillan, New York, 1959. MR**0103215 (21:1996)****[4]**JOHN McKAY & DAVID W. WALES, "The multipliers of the simple groups of order 604,800 and 50,232,960,"*J. Algebra*, v. 17, 1971, pp. 262-272. MR**0274577 (43:340)****[5]**CHARLES C. SIMS, "Determining the conjugacy classes of a permutation group," in*Computers in Algebra and Number Theory*(Proc. Sympos. Appl. Math., New York, 1970), SIAM-AMS Proc., Vol. 4, Amer. Math. Soc., Providence, R. I., 1971, pp. 191-195. MR**0338135 (49:2901)****[6]**CHARLES C. SIMS, "Computation with permutation groups," in*Proc. Second Sympos. Symbolic and Algebraic Manipulation*, Assoc. Comput. Mach., New York, 1971.**[7]**CHARLES C. SIMS, "Some algorithms based on coset enumeration," Unpublished notes, 1974.**[8]**CHARLES C. SIMS, "Some group theoretic algorithms," in*Topics in Algebra*, Lecture Notes in Math., Vol. 697, Springer-Verlag, Berlin, 1978. MR**524367 (81b:20026)****[9]**J. A. TODD, "Abstract definitions for the Mathieu groups,"*Quart. J. Math.*, v. 21, 1970, pp. 421-424. MR**0272878 (42:7759)****[10]**J. A. TODD & H. S. M. COXETER, "A practical method for enumerating cosets of a finite abstract group,"*Proc. Edinburgh Math. Soc.*, v. 5, 1936, pp. 26-34.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1980-0572868-5

Article copyright:
© Copyright 1980
American Mathematical Society