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.
