Arranging Countably Infinite Abelian Groups

An interesting result due to Nash-Williams (1959) can be used to construct enumerations of countably infinite abelian groups using sets of generators. In addition to providing a more accessible proof of this classical result, we compute, for any such group and any collection of generators, the cardinality of the set of such enumerations. These results can also be interpreted in terms of Cayley graphs for groups.