Let $G$ be a non-abelian group of order $2p$ for a prime $p>2$, $A, B \leq G$ s.t $|A|=p, |B|=2$, then $AB=G$
Looking for a simple proof that groups of order $2p$ are up to isomorphism $\mathbb{Z}_{2p}$ and $D_{p}$ .