Sylow p-Subgroups of Group of Order 2p/Proof 2

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Let $p$ be an odd prime.

Let $G$ be a group of order $2 p$.

Then $G$ has exactly one Sylow $p$-subgroup.

This Sylow $p$-subgroup is normal.


Let $n_p$ denote the number of Sylow $p$-subgroups of $G$.

From the First Sylow Theorem, there exists at least $1$ Sylow $p$-subgroup of $G$.

Let $P$ be such a Sylow $p$-subgroup of $G$.

The index of $P$ is $2$.

From Subgroup of Index 2 is Normal, $P$ is normal in $G$.

From Sylow $p$-Subgroup is Unique iff Normal it follows that there is only $1$ Sylow $p$-subgroup of $G$.