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Classification and non-classification results for abelian groups. (English) Zbl 0847.20050
Facchini, Alberto (ed.) et al., Abelian groups and modules. Proceedings of the Padova conference, Padova, Italy, June 23-July 1, 1994. Dordrecht: Kluwer Academic Publishers. Math. Appl., Dordr. 343, 135-143 (1995).
This fine paper surveys some results concerning (non-) classification of certain classes of abelian groups. Main objects are $$p$$-groups of cardinality $$\aleph_1$$, torsion-free groups of rank 2 and $$\aleph_1$$-separable groups of cardinality $$\aleph_1$$. The survey is based on a recent very general notion “classifiable” due to Melles (see the paper and references in this paper). One of the remarkable theorems (Melles) says: It is not provable in ZFC that the class of rank 2 torsion-free groups is classifiable. Sections 3 to 5 deal with uncountable groups. Results concerning Kaplansky’s test problems for various classes are discussed and new results using infinitary languages and Ehrenfeucht-Fraïssé games are considered from the present state of the art. The last section deals with $$\aleph_1$$-separable groups which attracted attention of the author and many others for more than a decade.
For the entire collection see [Zbl 0830.00031].
Reviewer: R.Göbel (Essen)
##### MSC:
 20K20 Torsion-free groups, infinite rank 20K10 Torsion groups, primary groups and generalized primary groups 03E35 Consistency and independence results 20K15 Torsion-free groups, finite rank 03C60 Model-theoretic algebra 03E25 Axiom of choice and related propositions