Group with One Sylow Subgroup per Prime Divisor is Solvable

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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.

$\blacksquare$


Sources