Hostname: page-component-848d4c4894-hfldf Total loading time: 0 Render date: 2024-05-15T02:51:02.156Z Has data issue: false hasContentIssue false
  • English
  • Français

Level and index in the modular group

Published online by Cambridge University Press:  14 November 2011

W. W. Stothers
W. W. Stothers
Affiliation:
Department of Mathematics, University of Glasgow, Glasgow G12 8QW
Get access Rights & Permissions [Opens in a new window]

Extract

It is shown that the index of a congruence subgroup of the modular group cannot be less than the level of the subgroup. This allows a number of existence theorems about non-congruence subgroups.

The level of a subgroup of the modular group can be defined in terms of the action on Q ∪ {∞}. We define a similar action to get information on congruence subgroups. In fact, we get a more powerful result, but this appears to be the most useful version.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 99 , Issue 1-2 , 1984 , pp. 115 - 126
Copyright
Copyright © Royal Society of Edinburgh 1984

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Brenner, J. L. and Lyndon, R. C.. Permutations and cubic graphs. Pacific J. Math. 104 (1983), 285–315.CrossRefGoogle Scholar
2Galois, E.. Letter to Chevalier. Liouville's J. 11 (1846), 381–444.Google Scholar
3Jones, G. A.. Triangular maps and non-congruence subgroups of the modular group. Bull. London Math. Soc. 11 (1979), 117–123.Google Scholar
4Larcher, H.. The cusp amplitudes of congruence subgroups of the classical modular group. Illinois J. Math. 26 (1982), 164–172.CrossRefGoogle Scholar
5Miller, G. A.. On the groups generated by two operators. Bull. Amer. Math. Soc. 7 (1901), 424–426.CrossRefGoogle Scholar
6Newman, M.. Bounds for the number of generators of a finite group. J. Res. Nat. Bur. Standards 71B (1967), 247–248.Google Scholar
7Newman, M.. Asymptotic formulas related to free products of cyclic groups. Math. Comput. 30 (1976) 838–846.CrossRefGoogle Scholar
8Rankin, R. A.. Modular functions and forms (Cambridge University Press, 1977).Google Scholar
9Stothers, W. W., Subgroups of the modular group. Proc. Cambridge Philos. Soc. 75 (1974), 139–153.Google Scholar
10Stothers, W. W.. Impossible specifications for the modular group. Manuscripta Math. 13 (1974), 415–428.CrossRefGoogle Scholar
11Stothers, W. W.. Subgroups of infinite index in the modular group. Glasgow Math. J. 19 (1978), 33–43.CrossRefGoogle Scholar
12Stothers, W. W.. Free subgroups of the free product of cyclic groups. Math. Comput. 32 (1978), 1274–1280.CrossRefGoogle Scholar
13Stothers, W. W.. Diagrams associated with subgroups of Fuchsian groups. Glasgow Math. J. 20 (1979), 103–114.CrossRefGoogle Scholar
14Stothers, W. W.. On a result of Petersson concerning the modular group. Proc. Roy. Soc. Edinburgh Sea. A 87 (1981), 263–270.CrossRefGoogle Scholar
15Wohlfahrt, K.. An extension of F. Klein's level concept. Illinois J. Math. 8 (1964), 529–535.Google Scholar