Let $|G|=2q$ for $q\geq3$ and $q$ is prime, and it has a normal subgroup of order $2$. Prove that $G$ is cyclic.
Here is what I did:
Let $D$ be a normal subgroup of $G$ and $|D|=2$, $D=\{e,a\}$ for some $a$. I found a quotient group of $|G/D|=q$. This implies that $G/D$ is cyclic therefore abelian.
But how does this imply that $G$ is cyclic? Because I know that a quotient group of cyclic is cyclic, but I don't know the other way around. Please help!
Thanks a lot!