Universal Abelian Groups

We examine the existence of universal elements in classes of infinite abelian groups. The main method is using group invariants which are defined relative to club guessing sequences. We prove, for example: Theorem: For n ≥ 2, there is a purely universal separable p-group in אn if, and only if, 20 ≤ אn. §0 Introduction In this paper “group” will always mean “infinite abelian group”, and “cardinal” and “cardinality” always refer to infinite cardinals and infinite cardinalities. Given a class of groups K and a cardinal λ we call a group G ∈ K universal for K in λ if |G| = λ and every H ∈ K with |H| ≤ λ 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 infinite abelian groups. We also investigate the existence of purely universal groups for K in λ, namely groups G ∈ K with |G| = λ such that every H ∈ K with |H| ≤ λ is isomorphic to a pure subgroup of G. 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 infinite object which are defined by looking at all possible enumerations of the object. Unlike the diamond and the square, two combinatorial principles which are already accepted as useful for the theory of infinite abelian groups, club guessing sequences are proved to exist in ZFC. Therefore using club guessing sequences does not require any additional axioms beyond the usual * Partially supported by the United States–Israel Binational science foundation. Publication number 455