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On the existence of normal maximal subgroups in division rings

Authors
Journal
Journal of Pure and Applied Algebra
0022-4049
Publisher
Elsevier
Publication Date
Volume
171
Identifiers
DOI: 10.1016/s0022-4049(01)00175-x
Disciplines
  • Mathematics

Abstract

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.

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