On matrix groups with finite spectrum

AbstractThe article deals with the question of what can we say about triangularizability of a group of matrices over a fieldF with characteristic zero under the assumption that the spectra of the elements of the group form a multiplicative subgroup of the fieldF. We restrict the problem on the case of finite spectrum and use the theory of algebraic groups to reduce the problem to the finite groups. The main result is an extension of Kolchin's theorem to the eigenvalues 1 and −1 followed by the counterexamples showing that this is the best possible extension under the mentioned assumptions