On Some Versions of Hopficity for Abelian Groups

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.