The following problem is from J.S. Milne book
Let $G$ be a finite Abelian group. If $G$ has at most $m$ elements of order dividing $m$ for each divisor of $m$ of $(G:1)$, show that $G$ is cyclic.
I do not understand the notation $(G:1)$. Also I would highly appreciate any hints (need not solve the problem as it coincides with my assignment) for solving the problem.