1. ## Cyclic groups

Let $G$ and $H$ be cyclic groups. If $|G| = m$ and $|H| = n$ and $(m,n) = 1$, show $G \times H$ is cyclic. (Hint: show $|(g,h)| = mn$.)

2. Suppose G=<g>, H=<h>, and |(g,h)|=k. Then $(e_G,e_H)=(g,h)^k=(g^k,h^k)$ so $g^k=e_G, h^k=e_H$. But then from the properties of the order, $m,n$ both divide $k$,
so k is a common multiple. Then what we want is the least common multiple of the orders, and that is $\dfrac{mn}{(m,n)}=mn=|G\times H|$ thus we have found the generator, and the result follows