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SUBGROUPS OF FINITE INDEX IN (2, 3, n)-TRIANGLE GROUPS

Published online by Cambridge University Press:  31 July 2012

W. WILSON STOTHERS
W. WILSON STOTHERS*
Affiliation:
Department of Mathematics, University Gardens, Glasgow, G12 8QW, Scotland
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Abstract

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For an integer n ≥ 7, let Δ(n) denote the (2, 3, n)-triangle group, and let M(n) be the positive integer determined by the conditions that Δ(n) has a subgroup of index m for all m ≥ M(n), but no subgroup of index M(n) − 1. The main purpose of the paper is to obtain information (bounds, in some cases explicit values) concerning the function M(n) (cf. Theorem 1). We also show that Δ(n) is replete (i.e., has a subgroup of index m for every integer m ≥ 1) if, and only if, n is divisible by 20 or by 30 (see Theorem 2).

Keywords

20D6020F67
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 54 , Issue 3 , September 2012 , pp. 693 - 714
Copyright
Copyright © Glasgow Mathematical Journal Trust 2012

References

REFERENCES

1.Conder, M. D. E., Generators for alternating and symmetric groups, J. London Math. Soc. 22 (1980), 75–86.CrossRefGoogle Scholar
2.Conder, M. D. E., More on generators for alternating and symmetric groups, Quart. J. Math. 32 (2) (1981), 137–163.CrossRefGoogle Scholar
3.Krattenthaler, C. and Müller, T. W., Parity patterns associated with lifts of Hecke groups, Abh. Math. Semin. Univ. Hambg. 78 (2008), 99–147.Google Scholar
4.Lehner, J., Discontinuous groups and automorphic functions, AMS Surveys No. 8 (American Mathematical Society, Providence, RI, 1964).CrossRefGoogle Scholar
5.Müller, T. W. and Schlage-Puchta, J.-C., Character theory of symmetric groups, subgroup growth of Fuchsian groups, and random walks, Adv. Math. 213 (2007), 919–982.CrossRefGoogle Scholar
6.Ree, R., A theorem on permutations, J. Comb. Theory Ser. A 10 (1971), 174–175.CrossRefGoogle Scholar
7.Singerman, D., Subgroups of Fuchsian groups and finite permutation groups, Bull. Lond. Math. Soc. 2 (1970), 319–323.CrossRefGoogle Scholar
8.Stothers, W. W., Impossible specifications for the modular group, Manuscripta Math. 13 (1974), 415–428.CrossRefGoogle Scholar
9.Stothers, W. W., Subgroups of the (2, 3, 7) triangle group, Manuscripta Math. 20 (1977), 323–334.Google Scholar
10.Stothers, W. W., Subgroups of (2, 3, n) triangle groups (unpublished).Google Scholar