From: rusin@vesuvius.math.niu.edu (Dave Rusin) Subject: Re: structure of arbitrary abelian groups Date: 30 Dec 1999 18:31:01 GMT Newsgroups: sci.math In article <19991230032003.13253.00002838@ng-fw1.aol.com>, FinalFntsy wrote: >Is it known if all abelian groups can be written (uniquely?) as a direct sum of >nonreducible abelian groups from some well specified set (including but >possibly not limited to Q, Z, Z_(p^k), p prime, the subset of Q consisting of >rational numbers with denominators powers of some fixed prime p, which I'll >call A_p, and A_p/Z)? Any proof almost definitely requires the Axiom of Choice >(or, as it would probably be applied, Zorn's Lemma), so I allow that it be >assumed. If this is not the case, could someone please provide a >counterexample (I think I remember reading about some abelian group G such that >G is isomorphic to G oplus G oplus G but not to G oplus G, and if so that would >probably be a counterexample). What's known is that "all Abelian groups" is much too murky a family of objects to permit this kind of structural theorem. If G is an abelian group then the collection of its elements of finite order forms a subgroup, its torsion subgroup T ; G/T is then torsion-free, But already the nicer result you would like to be true -- that G is in fact the direct sum of T and a torsion-free subsgroup H of G -- is false in general. An example of the failure would be the direct product (Z/2Z) x (Z/4Z) x (Z/8Z) x ... whose torsion subgroup is the direct _sum_ of the cyclic factors. These 'mixed' groups (for which T does not split) make it frustratingly difficult to provide good structure theorems. The torsion subgroup is in turn a direct sum of its p-primary subgroups. For these there is a reasonable structure theorem: one tallies up the cardinalities of various pieces to get what are called the Ulm invariants of the p-group. These completely characterize the group up to isomorphism _if_ the group is countable and contains no copy of what you call A_p/Z (the direct limit of the cyclic groups Z/(p^k Z).) I don't really think there is a good characterization of torsion-free abelian groups. Something as simple as the group of rational numbers already shows 'torsion-free' is more general than 'free'. You can find quite a bit of information in some classic books on the subject by Kaplansky and by Fuchs. I don't know of a more recent text offhand. Since there is no single nice structure theorem to render the subject trivial, you'll find that people focus on Abelian Groups in different ways: (1) An Abelian group is just a module for the ring of integers; rather than try to completely describe all modules for Z, it is more fruitful to find the most interesting results valid for modules over other rings (usually similar to Z in some way). (2) The subject should study not just the Abelian groups but also the maps (homomorphisms) between them; this is the Right Way according to category theory. This leads to the study of many extremely useful functors (Hom, Tor, Ext, ...) and opens up the field of Homological Algebra. (3) If the additive group of Q is interesting, why not look at the groups of other fields (of characteristic zero)? Well, they're all vector spaces over Q, and vector spaces are "trivial" (up to the axiom of choice so we can select a basis). But then again, no.[*] Indeed, the whole of Functional Analysis is the study of infinite-dimensional vector spaces, right? What makes Functional Analysis interesting is the presence of a metric, or topology, or order on the large spaces so as to give some structure to them. Likewise it is fruitful to study topological groups, or Abelian groups with some metric or order or other relation on them. (4) We have seen that torsion subgroups, at least, admit some characterization. It is similarly true that one can make some headway characterizing Abelian groups with some other limiting condition, such as divisible groups, groups of bounded height, matrix groups, etc. Of course there's more to do in a field than simply find a structure theorem for the objects. There are interesting questions about automorphism groups, representations, subgroup lattices, and so on. dave [*] If I was a sculptor, but then again no Or a man who makes potions in a traveling show... ============================================================================== From: Allan Adler Subject: Re: structure of arbitrary abelian groups Date: 30 Dec 1999 15:13:24 -0500 Newsgroups: sci.math My recollection could be faulty on this, but I vaguely recall that one of the striking parts of Shelah's proof of the independence of the Whitehead conjecture involves exhibiting an explicit exact sequence 0 -> Z -> A -> G -> 0 of abelian groups, where Z is the additive group of integers, such that it is independent of ZFC whether the sequence splits or not. I would expect that a result like this deals a death blow to any useful useful structure theory for arbitrary abelian groups. Allan Adler ara@zurich.ai.mit.edu **************************************************************************** * * * Disclaimer: I am a guest and *not* a member of the MIT Artificial * * Intelligence Lab. My actions and comments do not reflect * * in any way on MIT. Morever, I am nowhere near the Boston * * metropolitan area. * * * ****************************************************************************