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From the Wikipedia on abelian groups:

By contrast, classification of general infinitely-generated abelian groups is far from complete.

How far are we from a classification exactly? It seems like divisible groups have been classified. Which cases are left which we haven't? What is the nature of these unknown cases that makes them so hard to understand? Do we have examples of really "out there" infinitely-generated groups that don't fit into any of the current categories?

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up vote 13 down vote accepted

One problem with classification of objects of unrestricted size/complexity is set theoretic, and will be related to such strong set theoretic axioms related to accessible cardinals, measurable cardinals, and Vopenka's principle, and may depend on the continuum hypothesis and (certainly) by the axiom of choice. While I'm not an expert on these issues, I'll try to give a quick view of the difficulties.

For abelian groups, you can construct quite wild beasts using Zorn's Lemma. The general strategy will be to first construct some huge abelian group (say the product or coproduct of abelian groups indexed by a set of huge cardinality) and then apply Zorn's Lemma to obtain a maximal subgroup. Such subgroups typically can't be described directly and so would be very hard to study or classify.

Other ways of constructing very big objects is by taking ultraproducts of abelian groups. Here the construction relies on a choice of a principal ultrafilter, a highly non-constructive object. These objects are somewhat more tractable but still the immense freedom in choosing the cardinality of the indexing set, the ultrafilter, and the constituent groups makes this extremely hard.

Somewhat more systematically, since any abelian group is a quotient of a free abelian group, the complete classification requires the classification of the subgroups of free abelian groups. If no size restriction is placed on the number of generators of the free abelian groups again set theoretic subtleties immediately manifest themselves. For instance, taking a free abelian group on a set of huge cardinality, classifying the maximal subgroups in it will be very hard.

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Thanks, this is very interesting and helpful. I have a couple followup questions: 1. Is the challenge behind this type of counterexample generally going to be more set theoretic than group theoretic in nature? In particular, do examples exist of unclassified big abelian groups which are not problematic from a set theory standpoint? 2. Do these ever arise naturally (as groups) in other areas of math? Would classifying them be important enough to be worth the effort, assuming we could? – Samuel Handwich Feb 25 '13 at 21:26
A classification of abelian groups will entail a classification of the underlying sets of abelian groups, namely a classification of sets. So, the challenge will always include set theoretic components. Very large objects tend not to show up too often in the mainstream, for various philosophical and socio-mathematical reasons. I don't know of any particular interest in some truly big abelian group. I think it is currently too ambitious, and not sufficiently justified, to attempt such a classification. – Ittay Weiss Feb 25 '13 at 23:15

The book

L.Fuchs. Infinite Abelian Groups, Vol. I,II. Academic Press, 1970,1973

has a lot of unresolved problems. By them you can get an idea about actual directions in the theory of Abelian groups.

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And it is about an inch thick. So abelian groups are not as simple as they might first seem... – user1729 Feb 25 '13 at 13:06
Thanks. I looked this up at my library. There are also books with that title by Kaplansky and by Griffith. Do you have an opinion on those? – Samuel Handwich Feb 25 '13 at 21:32
To Samuel Handwich: I don't know, I didn't read it. – Boris Novikov Feb 25 '13 at 22:46

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