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Standard subgroups of GL2(A)

Published online by Cambridge University Press:  20 January 2009

A. W. Mason
A. W. Mason
Affiliation:
Department of Mathematics, University of Glasgow, Glasgow G12 8QW
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Let R be a commutative ring and let q be an ideal in R. Let En(R) be the subgroup of GLn(R) generated by the elementary matrices and let En(q) be the normal subgroup of En(R) generated by the q-elenientary matrices. The order of a subgroup S of GLn(R) is the ideal q0 in R generated by xij, xii−xjj, where (xij)∈S, with 1≦i, j≦n and i≠j. The subgroup S is called a standard subgroup if En(q0)≦S. An almost-normal subgroup of GLn(R) is a non-normal subgroup which is normalized by En(R).

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 30 , Issue 3 , October 1987 , pp. 341 - 349
Copyright
Copyright © Edinburgh Mathematical Society 1987

References

REFERENCES

1.Bass, H., Algebraic K-theory (Benjamin, New York, Amsterdam, 1968).Google Scholar
2.Bass, H., Milnor, J. and Serre, J-P., Solution of the congruence subgroup problem for SLn(n≧3) and SP2n(n≧2), Publ. Math. I.H.E.S. 33 (1967), 59–137.Google Scholar
3.Klingenberg, W., Lineare Gruppen über lokalen Ringen, Amer. J. Math. 83 (1961), 137–153.CrossRefGoogle Scholar
4.Liehl, B., On the group SL 2 over orders of arithmetic type, J. Reine Angew. Math. 323 (1981), 153–171.Google Scholar
5.Mason, A. W., On subgroups of GL(n, A) which are generated by commutators II, J. Reine Angew. Math. 322 (1981), 118–135.Google Scholar
6.Mason, A. W., Anomalous normal subgroups of the modular group, Comm. Algebra 11 (1983), 2555–2573.CrossRefGoogle Scholar
7.Mason, A. W., On non-normal subgroups of GLn(A) which are normalized by elementary matrices, Illinois J. Math. 28 (1984), 125–138.CrossRefGoogle Scholar
8.Mason, A. W., Anomalous normal subgroups of SL 2(K[x]), Quart. J. Math. (Oxford) Ser. (2), 36 (1985), 345–358.CrossRefGoogle Scholar
9.Mason, A. W., On GL 2 of a local ring in which 2 is not a unit, Canad. Math. Bull, to appear.Google Scholar
10.Mason, A. W. and Stothers, W. W., On subgroups of GL(n,A) which are generated by commutators, Invent. Math. 23 (1974), 327–346.Google Scholar
11.Newman, M., A complete description of the normal subgroups ofgenus one of the modular group, Amer. J. Math. 86 (1964), 17–24.Google Scholar
12.Ribenboim, P., Algebraic Numbers (Wiley-Interscience, New York, 1972).Google Scholar
13.Serre, J.-P., Le probléme des groupes de congruence pour SL 2, Ann. of Math. 92 (1970), 489–527.Google Scholar