It is the minimal separator in the sense that it corepresents the forgetful functor Grp -> Set, but this uses sets so probably isn't what you are after. In fact it is the same as the statement that Z is the free group on one generator.

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.