Co-Hopfian and boundedly endo-rigid mixed abelian groups

Mohsen Asgharzadeh, Mohammad Golshani and Saharon Shelah

Vol. 327 (2023), No. 2, 183–232

DOI: 10.2140/pjm.2023.327.183

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 $\mathrm{tor}<!--FUNCTION\; APPLICATION-->(G)=K$. 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 existence 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 $\lambda$.

Keywords

black boxes, bounded endomorphisms, co-Hopfian groups, endomorphism algebras, mixed abelian groups, $p$-groups, set theoretical methods in algebra

Mathematical Subject Classification

Primary: 03E75, 16S50, 20K30

Milestones

Received: 23 February 2023

Revised: 8 October 2023

Accepted: 6 November 2023

Published: 12 March 2024