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
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