Co-Hopfian and boundedly endo-rigid mixed abelian groups

TL;DR

This paper constructs boundedly endo-rigid abelian groups of specified sizes with particular torsion properties, characterizes pairs of torsion groups and cardinalities, and explores conditions for such groups to be co-Hopfian, especially in the mixed abelian case.

Contribution

It provides a complete characterization of pairs (K, λ) for boundedly endo-rigid abelian groups, introduces new classes of co-Hopfian groups, and offers criteria for co-Hopfianity in this context.

Findings

Constructed boundedly endo-rigid abelian groups with prescribed torsion.

Characterized pairs (K, λ) for such groups under mild conditions.

Identified cardinals where co-Hopfian abelian groups exist.

Abstract

For a given cardinal $\lambda$ and a torsion abelian group $K$ of cardinality less than $\lambda$, we present, under some mild conditions (for example $\lambda=\lambda^{\aleph_0}$), boundedly endo-rigid abelian group $G$ of cardinality $\lambda$ with $Tor(G)=K$. Essentially, we give a complete characterization of such pairs $(K, \lambda)$. Among other things, we use a twofold version of the black box. We present an application of the construction of boundedly endo-rigid abelian groups. Namely, we turn to the existing problem of co-Hopfian abelian groups of a given size, and present some new classes of them, mainly in the case of mixed abelian groups. In particular, we give useful criteria to detect when a boundedly endo-rigid abelian group is co-Hopfian and completely determine cardinals $\lambda> 2^{\aleph_{0}}$ for which there is a co-Hopfian abelian group of size $\lambda$.