Abelian Groups pp 343-408 | Cite as

*p*-Groups with Elements of Infinite Height

## Abstract

We continue our study of torsion groups concentrating on *p*-groups (with unspecified prime *p*) in the general case when the groups contain elements of infinite height. Matters are more subtle here as one has to deal with transfinite heights that are the central concept both in the search for invariants and in the proofs.

The focus of the structure theory is on *p*-groups that can be described by their UK-invariants. Accordingly, this chapter is primarily devoted to countable *p*-groups and their generalizations: the totally projective *p*-groups. The theory is perhaps the most interesting and highly satisfactory classification of a fairly large class of *p*-groups in terms of well-ordered sequences of cardinal numbers (provided by their UK-invariants). The four main approaches to the theory of totally projective *p*-groups (simple presentation, total projectivity, nice systems, and balanced-projectivity) underline the extreme importance of these groups; this theory is unparalleled in beauty and richness in abelian group theory.

Once the equivalence of the four main characterizations is established, there remain still some intriguing questions to be answered. For instance, which well-ordered sequences of cardinals may be the UK-invariants of a totally projective *p*-group? or, which is the largest class of *p*-groups that includes the generalized Prüfer groups, is closed under direct sums and summands, and whose members are distinguishable *via* their UK-invariants?

Needless to say, there have been various attempts to extend the well-rounded theory of totally projective *p*-groups, and various generalizations have been considered in the literature. So far these theories have produced only less remarkable results. Though several innovative techniques have been discovered, it seems that so far they have fallen short of true significance.

The final sections of this chapter deal with questions that are spin-offs of the theory of totally projective *p*-groups, and offer a glimpse into classes depending on ordinal numbers.