Sylow p-Subgroups of Group of Order 2p

Theorem
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.

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

From the Fourth Sylow Theorem:
 * $n_p \equiv 1 \pmod p$

From the Fifth Sylow Theorem:
 * $n_p \divides 2 p$

that is:
 * $n_p \in \set {1, 2, p, 2 p}$

But $p$ and $2 p$ are congruent to $0$ modulo $p$

So:
 * $n_p \notin \set {p, 2 p}$

Also we have that $p > 2$.

Hence:
 * $2 \not \equiv 1 \pmod p$

and so it must be that $n_p = 1$.

It follows from Normal Sylow P-Subgroup is Unique that this Sylow $p$-subgroup is normal.