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Non-Standard, Normal Subgroups and Non-Normal, Standard Subgroups of the Modular Group

Published online by Cambridge University Press:  20 November 2018

A.W. Mason
A.W. Mason*
Affiliation:
Department of Mathematics, University of Glasgow, Glasgow G12 8QW
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Abstract

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Let R be a commutative ring with identity. A subgroup S of GLn(R), where n ≥ 2, is said to be standard if and only if S contains all the q-elementary matrices and all conjugates of those matrices by products of elementary matrices, where q is the ideal in R generated by Xij,xii — Xjj(i ≠ j), for all (xij) ∊ S. It is known that, when n ≧ 3, the standard subgroups of GLn(R) are precisely those normalized by the elementary matrices. To demonstrate how completely this result can break down for n = 2 we prove that GL2(Z), where Z is the ring of rational integers, has uncountably many non-normal, standard subgroups and uncountably many non-standard, normal subgroups.

Keywords

20H05
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 32 , Issue 1 , 01 March 1989 , pp. 109 - 113
Copyright
Copyright © Canadian Mathematical Society 1989

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