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
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Normal and Composition Series: $\S 75 \epsilon$