Some of my friends and I were trying to discover a universal mapping property that characterizes the integers $\mathbb{Z}$ in the category of groups without referring to the underlying sets (So it is a no no to say it is the free group on a one element set). One of the big uses of $\mathbb{Z}$ is that it is a separator, i.e. for any two distinct pair of parallel arrows $f,g:A \rightarrow B$, there is at least one morphism $x:\mathbb{Z} \rightarrow A$ such that $f\circ x \neq g\circ x$. Unfortunately, any free group satisfies this property. I have two questions:

What are the separators in the category of groups (I think they will be just the free groups, but I have not proved this yet). Given this I think I can write down a universal property for Z which stays inside the category of groups.

Whether or not the above claim is correct, does anyone have a UMP that does the job?