On the existence of normal maximal subgroups in division rings

AbstractLet 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