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.