Two Generalizations of co-Hopfian Abelian Groups

TL;DR

This work investigates generalized co-Hopfian phenomena in Abelian groups by introducing the notions of relatively co-Hopfian and generalized co-Hopfian groups and positioning them relative to classical co-Hopfian and co-Bassian concepts. It proves a complete description for torsion-free generalized co-Hopfian groups: they are precisely the divisible groups; and for $p$-groups, a generalized co-Hopfian group is either divisible or splits as $G \cong A \oplus C$ with $C \cong \mathbb{Z}(p^n)^{(\kappa)}$ and $A$ co-Hopfian with $f_m(A)=0$ for all $m<n$. The authors further classify relatively co-Hopfian groups in the $p$-group setting, show that relatively co-Hopfian torsion groups are exactly the co-Hopfian ones, and establish comprehensive decompositional criteria for super and hereditarily relatively co-Hopfian groups, showing these two classes coincide. The paper also analyzes inheritance under direct sums and fully invariant subgroups, connects the theory to Ulm invariants and divisibility, and ends with open problems to stimulate further work. Throughout, all endomorphisms, Ulm invariants and decomposition structures are described with precise $p$-adic and module-theoretic language, providing a detailed map of the generalized co-Hopfian landscape for torsion, torsion-free, and mixed Abelian groups.

Abstract

By defining the classes of generalized co-Hopfian and relatively co-Hopfian groups, respectively, we consider two expanded versions of the generalized co-Bassian groups and of the classical co-Hopfian groups giving a close relationship with them. Concretely, we completely describe generalized co-Hopfian p-groups for some prime p obtaining that such a group is either divisible, or it splits into a direct sum of a special bounded group and a special co-Hopfian group. Furthermore, a comprehensive description of a torsion-free generalized co-Hopfian group is obtained. In addition, we fully characterize when a mixed splitting group and, in certain cases, when a genuinely mixed group are generalized co-Hopfian. Finally, complete characterizations of a super hereditarily generalized co-Hopfian group as well as of a hereditarily generalized co-Hopfian group are given, showing in the latter situation that it decomposes as the direct sum of three specific summands. Moreover, we totally classify relatively co-Hopfian p-groups proving the unexpected fact that they are exactly the co-Hopfian ones. About the torsion-free and mixed cases, we show in light of direct decompositions that in certain situations they are satisfactory classifiable -- e.g., the splitting mixed relatively co-Hopfian groups and the relatively co-Hopfian completely decomposable torsion-free groups. Finally, complete classifications of super and hereditarily relatively co-Hopfian groups are established in terms of ranks which rich us that these two classes curiously do coincide.

Theorems & Definitions (50)

• Lemma 2.1
• Proposition 2.2
• Proposition 2.3
• Proposition 2.4
• Lemma 2.5
• Proposition 2.6
• Proposition 2.7
• Corollary 2.8
• Corollary 2.9
• Corollary 2.10