Existence of Nonabelian Free Subgroups in the Maximal Subgroups of GLn(D)

Given a non-commutative finite dimensional F-central division algebra D, we study conditions under which every non-abelian maximal subgroup M of GL(n) (D) contains a non-cyclic free subgroup. In general, it is shown that either M contains a non-cyclic free subgroup or there exists a unique maximal subfield K of M,(D) such that N-GLn(D)(K*) = M, K* triangle M, K/F is Galois with Gal(K/F) congruent to M/K*, and F[M] = M-n(D). In particular, when F is global or local, it is proved that if ([D : F],Char(F)) = 1, then every nonabelian maximal subgroup of GL(1)(D) contains a non-cyclic free subgroup. Furthermore, it is also shown that GL(n)(F) contains no solvable maximal subgroups provided that F is local or global and n >= 5.