Suppose $H$ and $K$ are solvable subgroups of a group $G$ and $H \subseteq N_{G}(K)$. Prove that $HK$ is solvable.
I've tried to construct an abelian series of the form $HK \triangleright K \triangleright K_{1} \dots \triangleright K_{n}$, where the $K_{i}$ form an abelian series of $K$, but it doesn't work. Any ideas?
Hint: If $H^*$ is a subgroup of $H$, then $H^*K/H^*\simeq K/(H^*\cap K)$. Besides, since $K$ is solvable, every quotient of $K$ is solvable too.