Two generalizations of Hopfian Abelian groups

This paper targets to generalize the notion of Hopfian groups in the commutative case by defining the so-called relatively Hopfian groups and weakly Hopfian groups, respectively, and to establish some their crucial properties and characterizations. Specifically, we prove that for a reduced Abelian Formula: see text-group Formula: see text such that Formula: see text is Hopfian (in particular, is finite), the notions of relative Hopficity and ordinary Hopficity do coincide. We also show that if Formula: see text is a reduced Abelian Formula: see text-group such that Formula: see text is bounded and Formula: see text is Hopfian, then Formula: see text is relatively Hopfian. This allows us to construct a reduced relatively Hopfian Abelian Formula: see text-group Formula: see text with Formula: see text an infinite elementary group such that Formula: see text is not Hopfian. In contrast, for reduced torsion-free groups, we establish that the relative and ordinary Hopficity are equivalent. Moreover, the mixed case is explored as well, showing that the structure of both relatively and weakly Hopfian groups can be quite complicated.