Lie Groups, Representations and Discrete Mathematics

I will discuss a proof of the fact that given a finite dimensional division algebra D over an arbitrary field, any finite quotient of the multiplicative group D^* is solvable (joint work with Y.Segev and G.Seitz). Time permitting, I will also talk about some recent results that enable one to give a short proof of the Margulis-Platonov conjecture for the groups SL_{1,D}.