Summary
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.
Acknowledgment
The authors thank the referee for several constructive suggestions that improved the exposition in this work.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Additional information
Notes on contributors
Sylvie Corwin
SYLVIE CORWIN graduated from Whitman College in 2022 as a mathematics and English major. She explored Hamiltonian paths in abelian groups with the support of Patrick Keef and won that year’s “Best Senior Project” award. She currently lives in Seattle where she works in the actuarial field.
Patrick Keef
PATRICK KEEF (MR Author ID: 99710) is the Deshler Endowed Chair of Mathematics at Whitman College, where he has taught since 1980. His research has concentrated on abelian group theory, particularly primary groups, homological algebra and applications of logic and set theory.