Algebraic entropies, Hopficity and co-Hopficity of direct sums of Abelian Groups

Abstract

Necessary and sufficient conditions to ensure that the direct sum of two Abelian groups with zero entropy is again of zero entropy are still unknown; interestingly the same problem is also unresolved for direct sums of Hopfian and co-Hopfian groups.We obtain sufficient conditions in some situations by placing restrictions on the homomorphisms between the groups. There are clear similarities between the various cases but there is not a simple duality involved.

On the Existence of Uncountable Hopfian and co-Hopfian Abelian Groups

“…The research on Hopfian and co-Hopfian abelian groups has recently been revived thanks to its recently discovered connections with the study of algebraic entropy and its dual (see [6,12]), as e.g. groups of zero algebraic entropy are necessarily co-Hopfian (for more on the connections between these two topics see [13]). In this paper we will focus exclusively on abelian groups and for us "group" will mean "abelian group".…”

Generalised Hopficity and Products of the Integers

“…In particular, if G is R-Hopfian of the form G = F ⊕H, where F is free of infinite rank, then H must be torsion-free since otherwise G would have an R-Hopfian summand of the form F ⊕ C, where C is either finite or of the form Z(p ∞ ) for some prime p, both of which are impossible. Note a consequence of the above result: unlike the situation for Hopfian groups where the direct sum of a Hopfian group and a cyclic group is necessarily Hopfian (see [13] or [9]), the direct sum of a free (and hence R-Hopfian) group of infinite rank and a finite cyclic group is never R-Hopfian. In fact, this example shows that the class combinatorial arguments of Shelah's Black Box.…”

“…There is an extensive literature on both Hopfian and co-Hopfian groups, the papers [1,2,3,5,9,10,12,13,16,17] and the references therein give a small cross-section of the literature.…”

co-Hopfian Modules

“…Goldsmith and Gong [7] have a relaxed version of part (2) of the theorem:"If A, B are (co)-Hopfian and either Hom(A, B) = 0 or Hom(B, A) = 0, then A ⊕ B is (co)-Hopfian." This is a useful result in the case in which the torsion subgroup of an abelian group is a summand, and is a special case of a more general result they established in [5].…”