On a maximal torus in subgroups of a general linear group

Z. I. Borevich   ; A. A. Panin

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 4, Tome 227 (1995), pp. 15-22

Résumé

Let $k$ be a field, $K/k$ a finite extension of it of degree $n$. We denote $G=\operatorname{Aut}(_kK)$, $G_0=\operatorname{Aut}(_kK)$ and fix in $K$ a basis $\omega_1,\dots,\omega_n$ over $k$. In this basis, to any automorphism group of $_kK$ there corresponds a matrix group, which is denoted by the same symbol. Let $G'\le G$. In this paper, the conditions under which $G'\cap G_0$ is a maximal torus in $G'$ are studied. The calculation of $N_{G'}(G'\cap G_0)$ is carried out, provided that thee conditions are fulfilled. The case $G'=\operatorname{SL}(_kK)$ is of particular interset. It is known that for Galois extensions and for extensions of algebraic number fields, $G'\cap G_0$ is a maximal torus in $G'$. Bibligraphy: 2 titles.