Looking for a modern source about Ulm Invariants

Question

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.

Answer 1

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.