Exercise 17, § 5

Let K be an algebraically closed commutative field, V a finite-dimensional vector space over K, and G a submonoid of Aut(V) such that the K[G]-module V is simple. Prove that if the set of scalars Tr($g$), for $g\in G$, is finite, then G is finite (deduce from Burnside’s theorem that G contains a basis $(g_i)_{i\in I}$ of End(V), and consider the mapping $g\mapsto$ (Tr($gg_i$)) from G to $K^I$).