Let $A$ be a subset of natural numbers. Consider the following problem:
Is there a group $G$ such that $\lbrace O(x) \; | \; x \in G \rbrace = A\cup\lbrace 1\rbrace$ ? (where $O(x)$ is the order of $x$)
If you know any reference concerning this problem or any partial solution (containing a necessary or sufficient condition) , please let me know.