Let $G$ group of order $pq$ where $p,q$ primes.Show that if $G$ contains normal groups $N$ and $K$ with $|N|=p$ and $|K|=q$ then is cyclic Let $G$ group of order $pq$ where $p,q$ primes.Show that if $G$ contains normal groups $N$ and $K$ with $|N|=p$ and $|K|=q$ then is cyclic
Any ideas or hints for showing this?

Since $N$ and $K$ are normal, $NK$ is normal. Also $p,q$ are prime, so $N\cap K=1$. Since $NK\subset G$, by
$$
|NK|=\frac{|N||K|}{|N\cap K|}=pq=|G|
$$
we have $NK=G$. So $G\cong N\times K$. Since $|N|$ and $|K|$ are prime, $N$ and $K$ are cyclic, and thus $G$ is cyclic.
