The finite quotients of the multiplicative group of a division algebra of degree 3 are solvable

Let D be a finite dimensional division algebra. It is known that in a varie ty of cases, questions about the normal subgroup structure of D-x (the mult iplicative group of D) can be reduced to questions about finite quotients o f D-x. In this paper we prove that when deg(D) = 3, finite quotients of D-x are solvable. The proof uses Wedderburn's Factorization Theorem.