On matrix groups with finite spectrum

The article deals with the question of what can we say about triangularizab ility of a group of matrices over a field F with characteristic zero under the assumption that the spectra of the elements of the group form a multipl icative subgroup of the field F. We restrict the problem on the case of fin ite spectrum and use the theory of algebraic groups to reduce the problem t o the finite groups. The main result is an extension of Kolchin's theorem t o the eigenvalues 1 and -1 followed by the counterexamples showing that thi s is the best possible extension under the mentioned assumptions. (C) 1999 Elsevier Science Inc. All rights reserved.