Artinian groups of large cardinality

TL;DR

This paper explores the existence of large Artinian groups, linking the problem to universal algebras and the free subset problem, and discusses the consistency of positive or negative solutions under large cardinal assumptions.

Contribution

It establishes the equivalence of the Artinian group problem with questions in universal algebra and the free subset problem, providing insights into their consistent resolutions.

Findings

The problem is equivalent to an open question in universal algebra.

The problem is also equivalent to the free subset problem.

Both positive and negative solutions are consistent under different assumptions.

Abstract

A group is Artinian if there is no infinite strictly descending chain of subgroups. Ol'shanskii has asked whether there are Artinian groups of arbitrarily large cardinality. We show that this problem is essentially the same as an analogous question, regarding universal algebras, asked by J\'onsson in the 1960s. We further show that these problems are the same as the so-called free subset problem. As a result, one can have a consistent strong negative answer (from a large cardinal assumption) as well as a consistent positive answer.