On Some Versions of Hopficity for Abelian Groups

TL;DR

This paper characterizes certain classes of commutative co-finitely Hopfian groups and related variants, providing comprehensive descriptions and exploring their properties within the context of Abelian groups.

Contribution

It offers a complete description of co-finitely Hopfian groups in key cases and introduces and analyzes new variants like almost co-finitely Hopfian and almost finitely Hopfian groups.

Findings

Complete classification of co-finitely Hopfian groups in important cases

Introduction and analysis of almost co-finitely Hopfian groups

Discussion of more general Hopfian group variants

Abstract

We completely describe in certain important cases the class of commutative co-finitely Hopfian groups as defined by Bridson-Groves-Hillman- Martin in the journal Groups, Geometry, and Dynamics on 2010 (see [3]). We also consider and give a satisfactory description of several related classes of commutative groups. We also discuss in the commutative case a slightly more general version of co-finitely Hopfian groups called almost co-finitely Hopfian groups, as well as a more general version of Hopfian groups called almost finitely Hopfian groups.