摘要
Given a non-commutative finite dimensional F-central division algebra D, we study conditions under which every non-abelian maximal subgroup M of CLn (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 Mn(D) such that NCLn(D)(K*) = M, K* △ M, K/F is Galois with Gal(K/F) ≌ M/K*, and F[M] = in(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.
Given a non-commutative finite dimensional F-central division algebra D, we study conditions under which every non-abelian maximal subgroup M of CLn (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 Mn(D) such that NCLn(D)(K*) = M, K* △ M, K/F is Galois with Gal(K/F) ≌ M/K*, and F[M] = in(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.