On the existence of normal maximal subgroups in division rings

Abstract

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.