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 $D^{\ast }$ 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 $N$. In particular, we show that if either $N$ is algebraic over $F$ or $F$ is uncountable, then $N$ contains a non-abelian free subgroup.