OIMU 2008 Problem 7

Let $A$ be an abelian additive group such that all nonzero elements have infinite order and for each prime number $p$ we have the inequality $|A/pA|\leq p$ , where $pA = \{pa |a \in A\}$ , $pa = a+a+\cdots+a$ (where the sum has $p$ summands) and $|A/pA|$ is the order of the quotient group $A/pA$ (the index of the subgroup $pA$ ).

Prove that each subgroup of $A$ of finite index is isomorphic to $A$ .