A Note on the Multiplicative Group of a Division Ring

Let K be a noncommutative division ring with center Z and multiplicative group K*. Hua [2; 3] proved that (i) K*/Z* is a group without center, and (ii) K* is not solvable. A generalization (Theorem 1) will be given here which contains as a special case (Theorem 2) the fact that K*/Z* has no Abelian normal subgroups. This latter theorem obviously contains both (i) and (ii). As a further corollary it is shown that if M and N are normal subgroups of K* not contained in Z*, then MC\N is not contained in Z*. The final theorem is that an element x outside Z contains as many conjugates as there are elements in K. This makes more precise a theorem of Herstein [l], who showed that x has an infinite number of conjugates. Square brackets will denote multiplicative commutation. If 5 is a set, then o(S) will mean the number of elements in 5. A subgroup 77 of K* is subinvariant in K* if there is a chain {Ni} of subgroups such that H<\Nr-i< ■ ■ • <Ni<\K*, where A<B means that A is a normal subgroup of B.