Artinian groups of large cardinality

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.