Separators in the Category of Groups 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?
 A: 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.
A: Let $H$ be any group and let $G$ the direct product of $\mathbb{Z}$ and $H$. By sending all elements of $H$ to the identity, you can see that $G$ is also a separator.  It is easy to construct lots more examples along these lines (any group with $\mathbb{Z}$ as a quotient will do, for starters).
A: It is the minimal separator in the sense that it corepresents the forgetful functor Grp $\rightarrow$ Set, but this uses sets so probably isn't what you are after. In fact it is the same as the statement that $\mathbb{Z}$ is the free group on one generator.
A: A group surjects onto $\mathbb{Z}$ if and only if its abelianization tensored with the rationals is non-trivial.
A: In category theory there is the notion of a set $\mathcal S$ of generators for  a category $\mathcal C$. The defining property is that  for any morphisms $f,g:A \to B$ in $\mathcal C$ $f=g$ if and only $fa=ga$ for all morphisms $a:S \to A$ for all $S \in \mathcal S$. 
Thus for a set of generators morphisms with the same domain and codomain in $\mathcal C$ can be distinguished by morphisms from elements of $\mathcal S$. 
A more subtle property which is especially relevant to algebraic categories is that of density. A small subcategory $D$ of $\mathcal C$ is dense in $\mathcal C$ of for any objects $A,B$ of $\mathcal C$ the natural function 
$$\mathcal C(A,B) \to Nat_D(\mathcal C(-,A), \mathcal C(-,B))$$
which assigns to a morphism $f$ the natural transformation of functors $D\to \mathcal C$ which sends $\mathcal C(-,A) \to \mathcal C(-,B)$ by composition. Intuitively, this says that morphisms $A \to B$ can be recovered from morphisms from objects of $D$. 
A   dense subcategory  of the category of groups is that generated by the free groups on a 1 and 2 elements. 
For more discussion, see an article by Vaughan Pratt http://boole.stanford.edu/pub/yon.pdf. 
