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?



Separators in the Category of Groups
According to your definition, the separators will be exactly those groups $G$ with a surjection to $\mathbb{Z}$. One direction: if $f(x)\neq g(x)$, then take the composition $G \twoheadrightarrow \mathbb{Z} \rightarrow A$ where the latter map sends the generator of $\mathbb{Z}$ to $x$. For the other direction, to distinguish the maps $f,g: \mathbb{Z} \rightarrow \mathbb{Z}$ defined by $f(x)=x$ and $g(x)=2x$, your separator must surject to $\mathbb{Z}$.


This is a huge class of groups which has no particularly nice description beyond the definition, as far as I know.


