Geodesic
Non-Standard, Normal Subgroups and Non-Normal, Standard Subgroups of the Modular Group
Canadian mathematical bulletin, Tome 32 (1989) no. 1, pp. 109-113

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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 GL 2(Z), where Z is the ring of rational integers, has uncountably many non-normal, standard subgroups and uncountably many non-standard, normal subgroups.
DOI : 10.4153/CMB-1989-016-5
Mots-clés : 20H05 
Mason, A.W. Non-Standard, Normal Subgroups and Non-Normal, Standard Subgroups of the Modular Group. Canadian mathematical bulletin, Tome 32 (1989) no. 1, pp. 109-113. doi: 10.4153/CMB-1989-016-5
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     title = {Non-Standard, {Normal} {Subgroups} and {Non-Normal,} {Standard} {Subgroups} of the {Modular} {Group}},
     journal = {Canadian mathematical bulletin},
     pages = {109--113},
     year = {1989},
     volume = {32},
     number = {1},
     doi = {10.4153/CMB-1989-016-5},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1989-016-5/}
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