Elementary transvections in the overgroups of a non-split maximal torus

Dryaeva R. Y. , Koibaev, V. A.
Vladikavkaz Mathematical Journal 2015. Vol. 17. Issue 4.

Abstract:
A subgroup \(H\) of the general linear group \(GL(n, k)\) is rich in transvections if \(H\) contains elementary transvections \(t_{ij}(\alpha)\) at all positions \((i, j)\), \(i\neq j\). In this paper we show that if a subgroup \(H\) contains a non-split maximal torus and elementary transvection in one position, than \(H\) is rich in transvections. It is also proved that if a subgroup \(H\) contains a cyclic permutation of order \(n\) and elementary transvection at position \((i, j)\) such that numbers \(i-j\) and \(n\) are coprime, then \(H\) is rich in transvections.

Keywords: overgroup, intermediate subgroup, non-split maximal torus, transvection, elementary transvection

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