  We deal with the existence of universal members in a given cardinality for
several classes. First we deal with classes of Abelian groups, specifically
with the existence of universal members in cardinalities which are strong limit
singular of countable cofinality. Second, we then deal with (variants of) the
oak property (from a work of Dzamonja and the author), a property of complete
first order theories, sufficient for the non-existence of universal models
under suitable cardinal assumptions. Third, we prove that the oak property
holds for the class of groups (naturally interpreted, so for quantifier free
formulas) and deal more with the existence of universals.
