J\'onsson groups of various cardinalities

TL;DR

This paper establishes a connection between Jf3nsson groups and Jf3nsson algebras, showing that the existence of such groups depends on the same set-theoretic properties as algebras, and proves the existence of large Jf3nsson groups and confirms Babai's infinitary edge-orbit conjecture.

Contribution

It demonstrates the equivalence between the existence of Jf3nsson groups and Jf3nsson algebras across cardinalities, and proves the existence of arbitrarily large Jf3nsson groups.

Findings

Existence of Jf3nsson groups of any infinite cardinality.

Equivalence between Jf3nsson groups and Jf3nsson algebras.

Confirmation of Babai's infinitary edge-orbit conjecture.

Abstract

A group $G$ is J\'onsson if $|H| < |G|$ whenever $H$ is a proper subgroup of $G$. Using an embedding theorem of Obraztsov it is shown that there exists a J\'onsson group $G$ of infinite cardinality $\kappa$ if and only if there exists a J\'onsson algebra of cardinality $\kappa$. Thus the question as to which cardinals admit a J\'onsson group is wholly reduced to the well-studied question of which cardinals are not J\'onsson. As a consequence there exist J\'onsson groups of arbitrarily large cardinality. Another consequence is that the infinitary edge-orbit conjecture of Babai is true.