HK is cyclic if H and K are both cyclic Let $G$ be a finite group with normal subgroups $H$ and $K$ of relatively prime orders. Show that the group $HK$ is cyclic if $H$ and $K$ are both cyclic.
My attempt was to use the $2$nd Isomorphism Theorem, $H/(H \cap K)$ isomorphic to $HK/K$. Since $H$ and $K$ have relatively prime order, then $H \cap K={e}$, the trivial group. Therefore $H$ is isomorphic to $HK/K$. I am wondering how to apply normality to get $HK$ is cyclic if $H$ and $K$ are cyclic.

Sunce $H$ and $K$ are normal in $G$, they are normal in $HK$, and since they intersect trivially, $HK\cong H\times K$, and by the Chinese Remainder Theorem, the direct product of cyclic groups of relatively prime orders is cyclic, so $HK$ is cyclic.
