Some Classes of Hopfian Abelian Groups

n-Hopfian and n-co-Hopfian Groups

“…Suppose first that G is a Hopfian torsion-complete p-group. Assuming contradictiously that the basic subgroup B of such a group G is infinite, then would exist an epimorphism f : B → B which is not an automorphism (otherwise B will be a Hopfian direct sum of cyclic groups and thus by [7,Theorem 2] it must be finite, a contradiction). But it is well known that G = B, where B is the torsion completion of B in the p-adic topology (see [5]).…”

“…It is worthwhile noticing that the converse implication is not, however, true: Indeed, any infinite k-bounded group is not necessarily ω-co-Hopfian. We emphasize that Hopfian algebraically compact groups are described in ( [7], [8]). However, to the authors' knowledge, the complete description of co-Hopfian algebraically compact groups is not known to principally exist in the literature, so we offer a weaker version of it at the next statement.…”