Abstract
In a general linear group of degree 2 over field of rational numbersQ we consider subgroups containing a maximal non-split torus T=T(d) (that is, the image in G=GL(2,Q) of a multiplicative group of quadratic fieldQ(√d) under a regular imbedding). We investigate lattices Lat(d) depending on d of intermediate subgroups as well as structure of these subgroups. For any subgroup H from Lat(d) the factor-group Ng(H)/H is an Abelian group of exponent 2. The chain of successive normalizes H ≤NG(H) ≤NG(NG(H))≤... is stabilized in the final step. The dual descending chains H ≥TH ≥ T(t H) ≥...of successive normal closuresof torus T do not always terminate. A termination of such descending chains for all H from Lat(d) holds ifand only if d ≡ 1(rood 4). All the connected components of the graph of the normality relation on Lat(d)(garlands) are in a bijective correspondence to all the self-normalizing intermediate subgroups. We obtain adescription of all the self-normalizing and all the complete intermediate subgroups (F is complete if 7 FH = F).The proofs of the results are not given.
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Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademiya Nauk SSSR, Vol. 191, pp. 24–43, 1991.
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Borevich, Z.I., Koibaev, V.A. & Hoi, T.N. Lattices of subgroups inGL(2,Q) containing a non-split torus. J Math Sci 63, 622–633 (1993). https://doi.org/10.1007/BF01097974
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DOI: https://doi.org/10.1007/BF01097974