On a maximal torus in subgroups of a general linear group

Abstract

Let k be a field, K/k a finite extension of it of degree n. We denote G=Aut(kK), Go=Aut(k K) and fix in K a basis ω1,...,ωn 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′≤G., In this paper, the conditions under which G′⊎Go is a maximal torus in G′ are studied. The calculation of NG′(G′⊎Go) is carried out, provided that thee conditions are fulfilled. The case G′=SL (kK) is of particular interset. It is known that for Galois extensions and for extensions of algebraic number fields, G′⊎Go is a maximal torus in G′. Bibligraphy: 2 titles.