An abelian group always has a subgroup of order $\lcm(m,n)$

Consider the statements: (I) Let H and K be subgroups of an abelian group G. Suppose $|H|=m$ and $|K|=n$. Let $d=\lcm(m,n)$. Then G has a subgroup of order $d$. (II) The group of order 15 is abelian.