Universal Abelian

We examine the existence of universal elements in classes of innnite abelian groups. The method is deening some invariant of a group relative to a club guessing sequence, a combinatorial tool marketed here to algebraists. We prove, for example: Theorem: For n 2, there is a purely universal separable p-group in @ n if, and only if, 2 @ 0 @ n. + < < @ 0 , then there is no universal separable p-group of cardinality. x0 Introduction In this paper \group" will always mean \innnite abelian group", and \cardinal" and \cardinality" always refer to innnite cardinals and innnite cardinalities. We call a group G universal for a class of groups K in cardinality if every H 2 K such that jHj is isomorphic to a subgroup of G. The objective of this paper is to examine the existence of universal groups in various well-known classes of innnite abelian groups. We also investigate the existence of purely universal groups, groups with the property that every other group of equal cardinality in the class is isomorphic to one of its pure subgroups. The main set theoretic tool we use is a club guessing sequence. This is a prediction principle which has enough power to control properties of an innnite object which are deened by looking at all its possible enumerations. Unlike the diamond, club guessing sequences are proved to exist in ZFC. Therefore using them does not require any additional * Partially supported by the United States{Israel Binational science foundation. Publication number 455