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.
