Jónsson groups of various cardinalities

Abstract

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