Subgroup of Order p in Group of Order 2p is Normal

Theorem
Let $p$ be an odd prime.

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

Let $a \in G$ be of order $p$.

Let $K = \gen a$ be the subgroup of $G$ generated by $a$.

Then $K$ is normal in $G$.

Proof
The result Non-Abelian Order 2p Group has Order p Element demonstrates that such an element $a$ exists in $G$.

By definition of generator of cyclic group, $K$ is the cyclic group $C_p$ of order $p$.

By Lagrange's Theorem, the index of $K$ is:
 * $\index G K = \dfrac {\order G} {\order K} = \dfrac {2 p} p = 2$

The result follows from Subgroup of Index 2 is Normal.