On subnormal subgroups in division rings containing a non-abelian solvable subgroup

Let D be a division ring with center F and N a subnormal subgroup of the multiplicative group D* of D. Assume that N contains a non-abelian solvable subgroup. In this paper, we study the problem on the existence of non-abelian free subgroups in N. We show that if either N is algebraic over F or F is uncountable, then N contains a non-abelian free subgroup.