Group with One Sylow Subgroup per Prime Divisor is Solvable

Theorem
Let $G$ be a group of order $n$.

Suppose that, for each prime number $p$ which divides $n$, $G$ has exactly one $p$-Sylow subgroup.

Then $G$ is solvable.

Proof
From Finite Group with One Sylow p-Subgroup per Prime Divisor is Isomorphic to Direct Product,


 * $G$ is isomorphic to the direct product of its $p$-Sylow subgroups.

From Prime Power Group is Solvable, each $p$-Sylow subgroup is solvable.

From Direct Product of Solvable Groups is Solvable, $G$ is solvable.