On subgroups in division rings of type 2. (Q2859354)

A division ring over a field \(F\) is weakly locally finite (called here `of type \(2\)') if every subalgebra generated by two elements is finite. This class of division rings fits between algebraic and locally finite division rings (and is conjecturally equal to the latter). Notice that if every subalgebra generated by three elements is finite, then the algebra is locally finite, since a finite division ring is always generated by two elements.NEWLINENEWLINE Assume \(D\) is a weakly locally finite division ring. The authors show that \(F^\times\) is not contained in a finitely generated subgroup of \(D^\times\) (Theorem 2.5). They prove that in a noncentral subnormal subgroup \(N<D^\times\) there are \(x,y\in N\) such that \(y\) does not commute with any power of \(x\); and that any normal subgroup which is radical over some proper division subring is central (Theorem 3.2).