Submission: 2001, Jun 13
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 nontrivial 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.
2000 Mathematics Subject Classification: 16K20, 16K40
Keywords and Phrases: valuation, division ring, maximal subgroup
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