On the existence of normal maximal subgroups in division rings

✍ Scribed by S. Akbari; M. Mahdavi-Hezavehi

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 114 KB
• Volume: 171
• Category: Article
• ISSN: 0022-4049
• DOI: 10.1016/s0022-4049(01)00175-x

✦ Synopsis

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 ÿnite 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 ÿeld, 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.