Let G be an abelian group and let H and K be finite cyclic subgroups with |H| = r and |K| = s. Show that G contains a cyclic group of order lcm(r, s).
Let $G$ be abelian, $H$ and $K$ subgroups of orders $n$, $m$. Then G has subgroup of order $\operatorname{lcm}(n,m)$.