Geodesic
Elementary transvections in the overgroups of a non-split maximal torus
Vladikavkazskij matematičeskij žurnal, Tome 17 (2015) no. 4, pp. 11-17 
R. Y. Dryaeva; V. A. Koibaev. Elementary transvections in the overgroups of a non-split maximal torus. Vladikavkazskij matematičeskij žurnal, Tome 17 (2015) no. 4, pp. 11-17. http://geodesic.mathdoc.fr/item/VMJ_2015_17_4_a1/
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Borevich Z. I., “O podgruppakh lineinykh grupp, bogatykh transvektsiyami”, Zap. nauch. seminarov LOMI, 75, 1978, 22–31 | MR | Zbl

[2] Koibaev V. A., “Transvektsii v podgruppakh polnoi lineinoi gruppy, soderzhaschikh nerasschepimyi maksimalnyi tor”, Algebra i analiz, 21:5 (2009), 70–86 | MR | Zbl

[3] Koibaev V. A., Podgruppy gruppy $\mathrm{GL}(2,k)$, soderzhaschie nerasschepimyi tor, Itogi nauki. YuFU. Ser. mat. monografiya, 2, VNTs RAN i RSO-A, Vladikavkaz, 2009, 182 pp.