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 reduce this problem to an analogous question, regarding universal algebras, asked by J\'onsson in the 1960s. We provide Artinian groups of cardinality $\aleph_n$ for each natural number $n$. We also give a consistent strong negative answer to the question of Ol'shanskii (from a large cardinal assumption) as well as a consistent positive answer. Thus, answers to the questions of Ol'shanskii and J\'onsson are independent of set theory.