Degeneration and orbits of tuples and subgroups in an Abelian group

TL;DR

This paper investigates the structure of tuples and subgroups in countable reduced abelian groups, showing that degeneracy implies automorphism orbit equivalence and providing a detailed description of the orbit poset based on Ulm invariants.

Contribution

It establishes that degeneracy of tuples or subgroups in such groups implies they are in the same automorphism orbit and characterizes the orbit structure using Ulm invariants.

Findings

Degeneracy implies automorphism orbit equivalence in countable reduced abelian groups.

The orbit poset's structure depends solely on Ulm invariants.

A complete description of the orbit poset in terms of Ulm invariants is provided.

Abstract

A tuple (or subgroup) in a group is said to degenerate to another if the latter is an endomorphic image of the former. In a countable reduced abelian group, it is shown that if tuples (or finite subgroups) degenerate to each other, then they lie in the same automorphism orbit. The proof is based on techniques that were developed by Kaplansky and Mackey in order to give an elegant proof of Ulm's theorem. Similar results hold for reduced countably generated torsion modules over principal ideal domains. It is shown that the depth and the description of atoms of the resulting poset of orbits of tuples depend only on the Ulm invariants of the module in question (and not on the underlying ring). A complete description of the poset of orbits of elements in terms of the Ulm invariants of the module is given. The relationship between this description of orbits and a very different-looking one obtained by Dutta and Prasad for torsion modules of bounded order is explained.