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

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## Theorem

Let $G$ be a finite group whose order is $n$ and whose identity element is $e$.

Let $G$ be such that it has exactly $1$ Sylow $p$-subgroup for each prime divisor of $n$.

Then $G$ is isomorphic to the internal direct product of all its Sylow $p$-subgroups.

## Proof

If each of the Sylow $p$-subgroups are unique, they are all normal.

As the order of each one is coprime to each of the others, their intersection is $\set e$.

## Sources

- 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: The Sylow Theorems: $\S 59 \theta$