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I'm looking for a modern, approachable text (preferably a website, textbook, or expository article, and preferably one easily available online or at a library) which can explain the concept of Ulm invariants and how they factor into the classification of (countable) abelian groups.

The theorem I'm trying to understand is the following, apparently due to Mackey and Kaplansky:

A countable periodic reduced abelian group is determined uniquely up to isomorphism by its Ulm invariants for all prime numbers p and countable ordinals α.

Google covered the definitions of periodic (all elements of finite order; with abelian this is equivalent to locally finite) and reduced (no divisible elements but zero), but the definition of Ulm invariant is still a bit dodgy. The proof itself is not summarized, so it's unclear to what extent the following questions are answered or unanswered:

What if the group is not reduced? Do we have a characterization of (countable) divisible periodic abelian groups?

(apparently the divisible part is always a direct summand, so it's enough to do each part separately)

What if the group is not periodic? Can we use (e.g.) the finitely generated abelian groups theorem to sort of separate out the $\mathbb Z$-pieces?

What if the group is not countable? Can we see it canonically as made up of countable pieces, and assemble them?

I should say that I'm a logician, not an algebraist, so I don't know the "standard sources" in this field. But I'm looking to learn.

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  • $\begingroup$ Divisible abelian groups are injective in the category of abelian groups, so they always split. Every abelian group can be decomposed as $A=D\oplus R$, where $D$ is divisible and $R$ is reduced. Every divisible abelian group is isomorphic to a direct sum of copies of $\mathbb{Q}$ and copies of the Pr\"{u}fer group $\mathbf{Z}_{p^{\infty}}$, and two divisible groups are isomorphic if and only if they have the same cardinal of copies of each in this decomposition. This takes care of the divisible part of any abelian group, and answers your first question up to dealing with the reduced part. $\endgroup$ – Arturo Magidin Oct 23 '15 at 16:08
  • $\begingroup$ For torsion groups, we can separate them into their $p$-parts, and deal with the $p$-parts separately, and that is where the Ulm invariants come in. But for non-torsion groups, things get more dicey; you cannot in general just separate out the torsion and the non-torsion part, because the nontorsion part need not be free (like it is in the finitely generated case; because, for example, the direct product of $\omega$ copies of $\mathbb{Z}$ is not free). $\endgroup$ – Arturo Magidin Oct 23 '15 at 16:12
  • $\begingroup$ For uncountable groups, there is no easy decomposition as with the countable case, and the Ulm invariants do not completely determine the group. If you take the torsion subgroup of $\prod_{n=1}^{\infty}\mathbf{Z}_{p^n}$, this has the same Ulm invariants as the direct sum $\oplus_{n=1}^{\infty}\mathbf{Z}_{p^n}$, but is not isomorphic to it. There is no canonical decomposition in general into countable pieces. You can find a lot of the above done in Rotman's Introduction to the Theory of Groups 4th Ed. (Chapter 10); it mentions Ulm, but does not go into detail with them. $\endgroup$ – Arturo Magidin Oct 23 '15 at 16:14
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    $\begingroup$ Quid- I was posting those questions because they are my motivation for investigating this theorem, but I am primarily interested in a recommendation for a good source on this theorem, because I hope the proof is useful. And I'm just curious. Sorry for the confusion. $\endgroup$ – Richard Rast Oct 23 '15 at 16:24
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    $\begingroup$ If you are mainly after a general reference I'd second the recommendation of Fuchs' books in the comment above. The are quite standard and should not be hard to find. $\endgroup$ – user9072 Oct 23 '15 at 16:35
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A standard reference for abelian groups is Laszlo Fuch's books Infinite Abelian Groups. The Ulm-Kaplansky Theorem, characterising countable $p$-groups, is Theorem 77.3 on page 63 of Volume II. The Ulm sequence and Ulm type is defined on page 57; I'm only browsing it on-line, but it looks like it contains all relevant proofs.

That every abelian group can be written as $A=D\oplus R$ with $D$ divisible and $R$ reduced is fairly standard; it can be found in Rotman's Introduction to the Theory of Groups 4th Edition, Theorem 10.26, page 322. The fact that every divisible group is a direct sum of copies of $\mathbb{Q}$ and copies of Prüfer $p$-groups is Theorem 10.28, page 323. (If $D$ is divisible, then $D\cong \mathrm{tor}(D)\oplus V$, with $V$ torsion free and divisible; this means that $V$ is a vector space over $\mathbb{Q}$, hence a direct sum of copies of $\mathbb{Q}$. The $p$-primary component of $\mathrm{tor}(G)$ is divisible, and if you look at the subgroup of elements of exponent $p$ you get a vector space over $\mathbf{F}_p$, and that gives you the copies of the Prüfer group). The uniqueness of the decomposition is only stated and not proven, but it follows from Linear Algebra by considering suitable subgroups that determine the cardinality of the number of direct summands of the appropriate groups.

Rotman also has some results on torsion groups, and states (but does not prove) Ulm's Theorem. A couple of exercises show that the characterization via Ulm invariants does not extend to uncountable groups. The example suggested in the exercises is to consider $\mathrm{tor}(\prod_{n=1}^{\infty}\mathbf{Z}_{p^n})$ and $\oplus_{n=1}^{\infty}\mathbf{Z}_{p^n}$; the former is not a direct sum of cyclic groups.

In general you cannot separate a non-finitely generated mixed abelian group into "copies of $\mathbb{Z}$" and the rest in a satisfactory manner. Chapter XIV in volume II of Fuch's book discusses some results on mixed abelian groups.

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