Direct Product of Unique Sylow p-Subgroups is Unique Sylow p-Subgroup

Theorem

Let $G_1$ and $G_2$ be groups.

Let $H_1$ and $H_2$ be subgroups of $G_1$ and $G_2$ respectively.

Let $G_1$ be such that $H_1$ is the unique Sylow $p$-subgroup of $G_1$.

Let $G_2$ be such that $H_2$ is the unique Sylow $p$-subgroup of $G_2$.

Then $H_1 \times H_2$ is the unique Sylow $p$-subgroup of $G_1 \times G_2$.

Proof

From Direct Product of Sylow p-Subgroups is Sylow p-Subgroup, $H_1 \times H_2$ is a Sylow $p$-subgroup of $G_1 \times G_2$.

By Sylow $p$-Subgroup is Unique iff Normal, each of $H_1$ and $H_2$ are normal in $G_1$ and $G_2$ respectively.

By Direct Product of Normal Subgroups is Normal, $H_1 \times H_2$ is normal in $G_1 \times G_2$.

Again by Sylow $p$-Subgroup is Unique iff Normal, $H_1 \times H_2$ is the unique Sylow $p$-subgroup of $G_1 \times G_2$.

$\blacksquare$

Sources

• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $11$: The Sylow Theorems: Exercise $2$