حمیدرضا دربیدی

Given a non-commutative finite dimensional F-central division algebra D, we study conditions under which every non-abelian maximal subgroup M of GLn(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 sub¯eld K of Mn(D) such that N_GLn(D)(K^*)=M) M, K¤ CM, K=F is Galois with Gal(K=F»= M=K¤, and F[M] = Mn(D). In particular, when F is global or local, it is proved that if ([D : F]; Char(F)) = 1, then every non- abelian maximal subgroup of GL1(D) contains a non-cyclic free subgroup. Furthermore, it is also shown that GLn(F) contains no solvable maximal subgroups provided that F is local or global and n ¸ 5.