Let $A,B$ be solvable subgroup of a group $G$, suppose $B \subset N_G(A)$. Prove $AB$ is solvable Need help, Show: Let $A,B$ be solvable subgroup of a group $G$. Suppose the $B \subset N_G(A)$. Prove that $AB$ is solvable.

Because $B \subseteq N_G(A)$, it is easy to show that $AB$ is actually a subgroup. Since $B$ normalizes $A$ it follows that $A \unlhd AB$. Now $AB/A \cong B/(A \cap B)$ by the 2nd isomorphism theorem. Since $A$ is solvable and $B$, whence any of $B$'s quotients is solvable, it follows that $AB$ is solvable. (In general: $G$ is solvable iff $G/N$ and $N$ are solvable ($N \unlhd G$)).