Bassian-Finite Abelian Groups

TL;DR

This paper introduces the concept of Bassian-finite Abelian groups, exploring their properties and relationships with co-Hopfian and Dedekind-finite groups, including new constructions and extensions of existing results.

Contribution

It defines Bassian-finite groups, proves their equivalence with co-Hopfian groups in the torsion case, and constructs examples of Bassian-finite groups that are not co-Hopfian or not completely decomposable.

Findings

Bassian-finite groups coincide with co-Hopfian groups in the torsion case.

Constructed a torsion-free Bassian-finite group that is not co-Hopfian.

Found a countably infinite rank Butler group that is Bassian-finite but not completely decomposable.

Abstract

We introduce a new class of Abelian groups which lies strictly between the classes of co-Hopfian groups and Dedekind-finite groups, calling these groups {\it Bassian-finite}. We prove the surprising fact that in the torsion case the Bassian-finite property coincides with the co-Hopficity, thus extending a recent result by Chekhlov-Danchev-Keef in Siber. Math. J. (2026), and we construct a torsion-free Bassian-finite group which is {\it not} co-Hopfian as well as a Dedekind-finite group which is {\it not} Bassian-finite. Some other closely relevant things are also established. E.g., we extend a construction of a countable Butler group that is {\it not} completely decomposable, due to Arnold-Rangaswamy in Boll. Un. Mat. Ital. (2007), to find a Butler group of countably infinite rank which is Bassian-finite, but {\it not} completely decomposable.