Let $G$ be a finite group such that if $A, B\le G$ then $AB\le G$. Prove $G$ is a solvable.

Claim: All Sylow subgroups of $G$ are normal.

Proof: Suppose $P_1,P_2$ are different Sylow $p$-subgroups of $G$. Then the group $P_1P_2$ is a $p$-group that is strictly larger than $P_1$, contradiction.

Corollary: $G$ is solvable.