Jónsson groups of various cardinalities

Abstract

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