On the existence of normal maximal subgroups in division rings

Let D be a division ring with centre F. Denote by D* the multiplicative group of D. The relation between valuations on D and maximal subgroups of D* is investigated. In the finite dimensional case, it is shown that F* has a maximal subgroup if Br(F) is non-trivial provided that the characteristic of F is zero. It is also proved that if F is a local or an algebraic number field, then D* contains a maximal subgroup that is normal in D*. It should be observed that every maximal subgroup of D* contains either D′ or F*, and normal maximal subgroups of D* contain D′, whereas maximal subgroups of D* do not necessarily contain F*. It is then conjectured that the multiplicative group of any noncommutative division ring has a maximal subgroup.