Implementation and analysis of the Todd-Coxeter algorithm

Authors:
John J. Cannon, Lucien A. Dimino, George Havas and Jane M. Watson

Journal:
Math. Comp. **27** (1973), 463-490

MSC:
Primary 20-04

DOI:
https://doi.org/10.1090/S0025-5718-1973-0335610-5

MathSciNet review:
0335610

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A recent form of the Todd-Coxeter algorithm, known as the lookahead algorithm, is described. The time and space requirements for this algorithm are shown experimentally to be usually either equivalent or superior to the Felsch and Haselgrove-Leech-Trotter algorithms. Some findings from an experimental study of the behaviour of Todd-Coxeter programs in a variety of situations are given.

**[1]**M. J. Beetham, "A set of generators and relations for the groups ,*q*odd,"*J. London Math. Soc.*, v. 3, 1971, pp. 544-557. MR**44**#2806. MR**0285588 (44:2806)****[2]**A. Brunner, Personal communication.**[3]**C. M. Campbell, "Some examples using coset enumeration," in*Computational Problems in Abstract Algebra*(Edited by John Leech), Pergamon Press, New York, 1970, pp. 37-41. MR**40**#7341. MR**0254131 (40:7341)****[4]**H. S. M. Coxeter, "The abstract groups and ,"*Proc. London Math. Soc.*(2), v. 41, 1936, pp. 278-301.**[5]**H. S. M. Coxeter, "The abstract groups ,"*Trans. Amer. Math. Soc.*, v. 45, 1939, pp. 73-150. MR**1501984****[6a,b]**H. S. M. Coxeter & W. O. J. Moser,*Generators and Relations for Discrete Groups*, Springer-Verlag, Berlin, 1957; 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Band 14, Springer-Verlag, Berlin and New York, 1965. MR**19**, 527; MR**30**#4818. MR**0088489 (19:527d)****[7]**L. A. Dimino, "A graphical approach to coset enumeration,"*SIGSAM Bulletin*, v. 19, 1971, pp. 8-43.**[8]**H. Felsch, "Programmierung der Restklassenabzählung einer Gruppe nach Untergruppen,"*Numer. Math.*, v. 3, 1961, pp. 250-256. MR**24**#B488. MR**0134435 (24:B488)****[9]**M. J. T. Guy,*Coset Enumeration*, Lecture delivered at the Conference on Computational Problems in Abstract Algebra, Oxford, 29 August-2 September, 1967.**[10]**G. Higman & J. McKay, "On Janko's simple group of order 50,232,960,"*Bull. London Math. Soc.*, v. 1, 1969, pp. 89-94. MR**40**#224. MR**0246955 (40:224)****[11]**J. Leech, "Coset enumeration on digital computers,"*Proc. Cambridge Philos. Soc.*, v. 59, 1963, pp. 257-267. MR**26**#4513. MR**0146994 (26:4513)****[12]**J. Leech, "Generators for certain normal subgroups of (2,3,7),"*Proc. Cambridge Philos. Soc.*, v. 61, 1965, pp. 321-332. MR**30**#4821. MR**0174621 (30:4821)****[13]**J. Leech, "Coset enumeration," in*Computational Problems in Abstract Algebra*(Edited by John Leech), Pergamon Press, New York and Oxford, 1970, pp. 21-35. MR**40**#7343. MR**0254133 (40:7343)****[14]**J. Leech & J. Mennicke, "Note on a conjecture of Coxeter,"*Proc. Glasgow Math. Assoc.*, v. 5, 1961, pp. 25-29. MR**27**#196. MR**0150193 (27:196)****[15]**I. D. Macdonald, "On a class of finitely presented groups,"*Canad. J. Math.*, v. 14, 1962, pp. 602-613. MR**25**#3992. MR**0140574 (25:3992)****[16]**J. McKay & D. Wales, "The multipliers of the simple groups of order 604,800 and 50,232,960,"*J. Algebra*, v. 17, 1971, pp. 262-272. MR**43**#340. MR**0274577 (43:340)****[17]**N. S. Mendelsohn, "An algorithmic solution for a word problem in group theory,"*Canad. J. Math.*, v. 16, 1964, pp. 509-516; Correction,*Canad. J. Math.*, v. 17, 1965, p. 505. MR**29**#1248;**31**#237. MR**0163949 (29:1248)****[18]**J. L. Mennicke & D. Garbe, "Some remarks on the Mathieu groups,"*Canad. Math. Bull.*, v. 7, 1964, pp. 201-212. MR**29**#150. MR**0162846 (29:150)****[19]**B. H. Neumann, Personal communication.**[20]**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.**[21]**H. F. Trotter, "A machine program for coset enumeration,"*Canad. Math. Bull.*, v. 7, 1964, pp. 357-368; Program listing in*Mathematical Algorithms*, v. 1, 1966, pp. 12-18. MR**29**#5415. MR**0168151 (29:5415)****[22]**J. M. Watson,*An Extended Todd-Coxeter Algorithm*, Department of Mathematics, Institute of Advanced Studies, Australian National University, Canberra, 1971, 15 pages.

Retrieve articles in *Mathematics of Computation*
with MSC:
20-04

Retrieve articles in all journals with MSC: 20-04

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1973-0335610-5

Keywords:
Todd-Coxeter algorithm,
generators and relations,
group theory

Article copyright:
© Copyright 1973
American Mathematical Society