On subgroups of the general linear group that contain a maximal nonsplit torus.

The paper deals with the structure of intermediate subgroups of the general linear group GL( n, k) of degree n over a field k of odd characteristic that contain a nonsplit maximal torus related to a radical extension of degree n of the ground field k. The structure of ideal nets over a ring that determine the structure of intermediate subgroups containinga transvection is given. Let $$ K = k\left( {\sqrt[n]{d}} \right) $$ be a radical degree-n extension of a field k of odd characteristic, and let T =( d) be a nonsplit maximal torus, which is the image of the multiplicative group of the field K under the regular embedding in G =GL( n, k). In the paper, the structure of intermediate subgroups H, T ≤ H ≤ G, that contain a transvection is studied. The elements of the matrices in the torus T = T ( d) generate a subring R( d) in the field k.Let R be an intermediate subring, R( d) ⊆ R ⊆ k, d ∈ R. Let σ denote the net in which the ideal dR stands on the principal diagonal and above it and all entries of which beneath the principal diagonal are equal to R. Let σ denote the net in which all positions on the principal diagonal and beneath it are occupied by R and all entries above the principal diagonal are equal to dR. Let E(σ) be the subgroup generated by all transvections from the net group G(σ). In the paper it is proved that the product TE(σ) is a group (and thus an intermediate subgroup). If the net σ associated with an intermediate subgroup H coincides with σ,then TE(σ) ≤ H ≤ N(σ),where N(σ) is the normalizer of the elementary net group E(σ) in G. For the normalizer N(σ),the formula N(σ)= TG(σ) holds. In particular, this result enables one to describe the maximal intermediate subgroups. Bibliography: 13 titles. [ABSTRACT FROM AUTHOR]