Free Subgroups in Maximal Subgroups of GL1(D)

Let D be a division algebra of finite dimension over its center F. Given a noncommutative maximal subgroup M of D* GL1(D), it is proved that either M contains a noncyclic free subgroup or there exists a maximal subfield K of D which is Galois over F such that K* is normal in M and M/K* Gal(K/F). Using this result, it is shown in particular that if D is a noncrossed product division algebra, then M does not satisfy any group identity.