Applications and adaptations of the low index subgroups procedure

Conder, Marston; Dobcsányi, Peter

doi:10.1090/S0025-5718-04-01647-3

Applications and adaptations of the low index subgroups procedure
HTML articles powered by AMS MathViewer

by Marston Conder and Peter Dobcsányi PDF

Math. Comp. 74 (2005), 485-497 Request permission

Abstract:

The low-index subgroups procedure is an algorithm for finding all subgroups of up to a given index in a finitely presented group $G$ and hence for determining all transitive permutation representations of $G$ of small degree. A number of significant applications of this algorithm are discussed, in particular to the construction of graphs and surfaces with large automorphism groups. Furthermore, three useful adaptations of the procedure are described, along with parallelisation of the algorithm. In particular, one adaptation finds all complements of a given finite subgroup (in certain contexts), and another finds all normal subgroups of small index in the group $G$. Significant recent applications of these are also described in some detail.

References

W. Bosma and J. J. Cannon, Handbook of Magma Functions, University of Sydney, 1996.
John J. Cannon, Lucien A. Dimino, George Havas, and Jane M. Watson, Implementation and analysis of the Todd-Coxeter algorithm, Math. Comp. 27 (1973), 463–490. MR 335610, DOI 10.1090/S0025-5718-1973-0335610-5
Marston Conder, An infinite family of $5$-arc-transitive cubic graphs, Ars Combin. 25 (1988), no. A, 95–108. Eleventh British Combinatorial Conference (London, 1987). MR 942495
Marston Conder, The symmetric genus of the Mathieu groups, Bull. London Math. Soc. 23 (1991), no. 5, 445–453. MR 1141014, DOI 10.1112/blms/23.5.445
Marston Conder, Asymmetric combinatorially-regular maps, J. Algebraic Combin. 5 (1996), no. 4, 323–328. MR 1406456, DOI 10.1023/A:1022496532169
M. D. E. Conder and P. Dobcsányi, Normal subgroups of low index in the modular group and other Hecke groups, University of Auckland Mathematics Department Research Report Series, No. 496, 23 pp., 2003.
Marston Conder and Peter Dobcsányi, Determination of all regular maps of small genus, J. Combin. Theory Ser. B 81 (2001), no. 2, 224–242. MR 1814906, DOI 10.1006/jctb.2000.2008
Marston Conder and Peter Dobcsányi, Trivalent symmetric graphs on up to 768 vertices, J. Combin. Math. Combin. Comput. 40 (2002), 41–63. MR 1887966
Marston Conder and Brent Everitt, Regular maps on non-orientable surfaces, Geom. Dedicata 56 (1995), no. 2, 209–219. MR 1338960, DOI 10.1007/BF01267644
Marston Conder and Peter Lorimer, Automorphism groups of symmetric graphs of valency $3$, J. Combin. Theory Ser. B 47 (1989), no. 1, 60–72. MR 1007714, DOI 10.1016/0095-8956(89)90065-8
M. D. E. Conder, C. Maclachlan, S. Todorovic-Vasiljevic and S. E. Wilson, Bounds for the number of automorphisms of a compact non-orientable surface, J. London Math. Soc. 68 (2003), 65–82.
Marston D. E. Conder and Gaven J. Martin, Cusps, triangle groups and hyperbolic $3$-folds, J. Austral. Math. Soc. Ser. A 55 (1993), no. 2, 149–182. MR 1232754
H. S. M. Coxeter and W. O. J. Moser, Generators and relations for discrete groups, 4th ed., Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 14, Springer-Verlag, Berlin-New York, 1980. MR 562913
Anke Dietze and Mary Schaps, Determining subgroups of a given finite index in a finitely presented group, Canadian J. Math. 26 (1974), 769–782. MR 407159, DOI 10.4153/CJM-1974-072-0
P. Dobcsányi, Adaptations, Parallelisation and Applications of the Low-index Subgroups Algorithm, PhD thesis, University of Auckland, 142 pp., 1999.
P. Dobcsányi, home page, http://www.math.auckland.ac.nz/˜peter.
B. J. Everitt, Images of Hyperbolic Reflection Groups, PhD thesis, University of Auckland, 153 pp., 1994.
George Havas, A Reidemeister-Schreier program, Proceedings of the Second International Conference on the Theory of Groups (Australian Nat. Univ., Canberra, 1973) Lecture Notes in Math., Vol. 372, Springer, Berlin, 1974, pp. 347–356. MR 0376827
D. F. Holt and W. Plesken, A cohomological criterion for a finitely presented group to be infinite, J. London Math. Soc. (2) 45 (1992), no. 3, 469–480. MR 1180256, DOI 10.1112/jlms/s2-45.3.469
D. L. Johnson, Presentations of groups, London Mathematical Society Student Texts, vol. 15, Cambridge University Press, Cambridge, 1990. MR 1056695
Peter Lorimer, Hyperbolic pyritohedra constructed from the Coxeter group $[4,3,5]$, Computational algebra and number theory (Sydney, 1992) Math. Appl., vol. 325, Kluwer Acad. Publ., Dordrecht, 1995, pp. 303–321. MR 1344939
J. Neubüser, An elementary introduction to coset table methods in computational group theory, Groups—St. Andrews 1981 (St. Andrews, 1981) London Math. Soc. Lecture Note Ser., vol. 71, Cambridge Univ. Press, Cambridge-New York, 1982, pp. 1–45. MR 679153
Charles C. Sims, Computation with finitely presented groups, Encyclopedia of Mathematics and its Applications, vol. 48, Cambridge University Press, Cambridge, 1994. MR 1267733, DOI 10.1017/CBO9780511574702