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The order and level of a subgroup of GL2 over a Dedekind ring of arithmetic type

Published online by Cambridge University Press:  14 November 2011

A. W. Mason
A. W. Mason
Affiliation:
Department of Mathematics, University of Glasgow, University Gardens, Glasgow G12 8QW, Scotland, U.K.
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Synopsis

Let R be a commutative ring and let q be an R-ideal. Let En(R) be the subgroup of GLn(R) generated by the elementary matrices and let En(R, q) be the normal subgroup of En(R) generated by the q-elementary matrices. For each subgroup S of GLn(R) the order of S, o(S), is the R-ideal generated by xij, xii − xjj (i ≠ j), where (xij) ∈ S, and the level of S, l(S), is the largest R-ideal q0 with the property that En (R, q0) ≦ S. It is known that when n ≧ 3, the subgroup S is normalised by En(R) if and only if o(S) = l(S). It is also known that this result does not hold when n = 2. For example, there are uncountably many normal subgroups S of SL2(ℤ) such that o(S) ≠ {0} and l(S) = {0}, where ℤ is the ring of integers. In this paper we prove that, when A is a Dedekind ring of arithmetic type containing infinitely many units, the order q and level q′ of a subgroup S of GL2(A), normalised by E2(A), are closely related. It is proved that Ψ(q)≦q′, where ≦(q) = 12uq, with u the A-ideal generated by u2 − 1 (u ∈ A*), when A is contained in a number field, and Ψ(q) = q3, when A is contained in a function field.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 119 , Issue 3-4 , 1991 , pp. 191 - 212
Copyright
Copyright © Royal Society of Edinburgh 1991

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References

1Bass, H.. Algebraic K-theory (New York: Benjamin, 1968).Google Scholar
2Bass, H., Milnor, J. and Serre, J.-P.. Solution of the congruence subgroup problem for SL 2n (n ≧ 3) and Sp 2n (n ≧ 2). Publ. Math. I.H.E.S. 33 (1967), 59–137.CrossRefGoogle Scholar
3Beyl, F. R.. The Schur Multiplicator of SL(2, ℤ/mℤ) and the Congruence Subgroup Property, Math. Z. 191 (1986), 23–42.CrossRefGoogle Scholar
4Costa, D. L. and Keller, G. E.. On the normal subgroups of SL(2, A), J. Pure Appl. Algebra 53 (1988), 201–226.CrossRefGoogle Scholar
5Klingenberg, W.. Lineare Gruppen über lokalen Ringen. Amer. J. Math. 83 (1961), 137–153.CrossRefGoogle Scholar
6Lacroix, N. H. J.. Two dimensional linear groups over local rings. Canad. J. Math. 21 (1969), 106–135.CrossRefGoogle Scholar
7Liehl, B.. On the group SL2 over orders of arithmetic type. J. Reine Angew. Math. 323 (1981), 153–171.Google Scholar
8McQuillan, D. L.. Classification of normal congruence subgroups of the modular group. Amer. J. Math. 87 (1965), 285–296.CrossRefGoogle Scholar
9Mason, A. W.. Anomalous normal subgroups of the modular group. Comm. Algebra 11 (1983), 2555–2573.CrossRefGoogle Scholar
10Mason, A. W.. Standard subgroups of GL 2(A). Proc. Edinburgh Math. Soc. 30 (1987), 341–349.CrossRefGoogle Scholar
11Mason, A. W.. On GL2 of a local ring in which 2 is not a unit. Canad. Math. Bull. 30 (1987), 165–176.CrossRefGoogle Scholar
12Mason, A. W.. Free quotients of congruence subgroups of SL2 over a coordinate ring. Math. Z. 198 (1988), 39–51.CrossRefGoogle Scholar
13Mason, A. W.. Non-standard, normal subgroups and non-normal, standard subgroups of the modular group. Canad. Math. Bull. 32 (1989), 109–113.CrossRefGoogle Scholar
14Mason, A. W.. On GL2 of a local ring in which 2 is not a unit II. Comm. Algebra. 17 (1989), 511–551.CrossRefGoogle Scholar
15Mason, A. W. and Stothers, W. W.. On subgroups of GL(n, A) which are generated by commutators. Invent. Math. 23 (1974), 327–346.CrossRefGoogle Scholar
16Ribenboim, P. R.. Algebraic Numbers (New York: Wiley-Interscience, 1972).Google Scholar
17Serre, J.-P.. Le problème des groupes de congruence pour SL2. Ann. of Math. 92 (1970), 489–527.CrossRefGoogle Scholar
18Vaserstein, L. N.. On the group SL2 over Dedekind rings of arithmetic type. Math. USSR-Sb. 18 (1972), 321–332.CrossRefGoogle Scholar
19Vaserstein, L. N.. On the normal subgroups of GLn over a ring. In Lecture Notes in Mathematics, 854, pp. 456–465. (Berlin-New York: Springer, 1981).Google Scholar
20Vaserstein, L. N.. Normal subgroups of the general linear groups over Banach algebras. J. Pure Appl. Algebra 41 (1986), 99–112.CrossRefGoogle Scholar