Abelian Groups pp 299-342 | Cite as

# Torsion Groups

## Abstract

We are now prepared to plunge into an in-depth study of the major classes of abelian groups. Divisible groups have been fully characterized, so we can concentrate on reduced groups. Our discussion begins with the theory of torsion groups. Since a torsion group is a direct sum of uniquely determined *p*-groups, it is clear that the study of torsion groups reduces immediately to *p*-groups. This chapter is primarily concerned with *p*-groups without elements of infinite heights (called separable *p*-groups), while the next chapter will concentrate on *p*-groups containing elements of infinite heights.

Separable *p*-groups are distinguished by the property that every finite set of elements is contained in a finite summand. This proves to be a very powerful property. However, as it turns out, it does not simplify the group structure to the extent one hopes for: though the full potential of this condition has not been realized, it looks probable (if not certain) that a reasonable classification in terms of the available invariants is impossible. Every separable *p*-group is a pure subgroup between its basic subgroup *B* and the largest separable *p*-group \(\overline{B}\) with the same basic subgroup. Much of the interest in these torsion-complete *p*-groups \(\overline{B}\) comes from their numerous remarkable algebraic and topological features, one of which is that they admit complete systems of invariants.

There is a large body of work available on separable *p*-groups, and a fair amount of material will be covered on such groups that are distinguished by interesting properties, mostly those shared by torsion-complete groups.