Purity and Basic Subgroups
In this chapter, we are going to discuss a basic concept: pure subgroup. This concept has been one of the most fertile notions in the theory since its inception in a paper by the pioneer H. Prüfer. The relevance of purity in abelian group theory, and later in module theory, has tremendously grown with time. While abelian groups have been major motivation for a number of theorems in category theory, purity has served as a prototype for relative homological algebra, and has played a significant role in model theory as well.
Pure subgroups, and their localized version: p-pure subgroups, are often used as a weakened notion of summands. In contrast to summands, most groups admit a sufficient supply of pure subgroups: every infinite set of elements embeds in a pure subgroup of the same cardinality. They are instrumental in several results that furnish us with criteria for a summand.
Every group contains, for every prime p, a p-pure subgroup, called p-basic subgroup, that is (if not zero) a direct sum of infinite cyclic groups and cyclic p-groups. Basic subgroups are unique up to isomorphism, and store relevant information about the containing group. Basic subgroups were introduced by Kulikov for p-groups, and occupy a center stage in the theory of these groups.