Explicit example of Abelian group with non-zero ulm invariants for infinite ordinals

Question

I have seen a few posts here about Ulm invariants of infinite Abelian groups, and I know that countable Abelian groups are classified up to isomorphism by their Ulm invariants (and I think any combination of Ulm invariants is possible?). If I recall correctly, the length of the group (supremum of ordinals such that Ulm invariants are non-zero) can be any countable ordinal. However, I don't have a sense of what a group with infinite Ulm invariants should look like, I haven't been able to puzzle it out on my own, and I haven't been able to find any source that gives an explicit example of one.
I would appreciate it if someone could provide me with one (or multiple) explicit constructions of a reduced Abelian p-group with elements of infinite height, and ideally a way to construct one of any given length.

Answer 1

Any ordinal can be the length of a reduced abelian p-group. A construction by generators and relations of a simply presented group whose length is a successor ordinal was given in E. A. Walker, The groups $P_\beta$, Symposia Math. (13), 482-488, (1974); the construction is presented on p. 356, L. Fuchs, Abelian Groups, (2015). For limit ordinals, just take the direct limit of a direct system of groups of smaller length. These examples probably occur in earlier editions of Fuchs, but I don't know where.

Here is an explicit example due to Pruefer, of a countable p--group $G$ of length $\omega + 1$. Let $a_0, a_1, \dots$ be a countable set of generators with relations $pa_0=0,\ pa_1=a_0,\dots, p^na_n=a_0,\dots$. Then $a_0$ has order $p$ and height $\omega$ and for all $n$, $a_n$ has order $p^{n+1}$ and height $0$.