Abelian Groups - László Fuchs [Springer]

Chapter 5
Purity and Basic Subgroups
Abstract 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,ap-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.
1 Purity
Pure Subgroups A subgroup G of a group A is called pure if the equation
nx D g 2 G .n 2 N/ is solvable for x in G whenever it is solvable in A. This
amounts to saying that G is pure in A if, for any g 2 G, njg in A implies njg in G.
As njg in G means g 2 nG, we see that G is pure in A if and only if
nG D G \ nA for every n 2 N: (5.1)
Thus purity means that the divisibility properties of the elements in G by integers
are the same whether computed in A or in G.
If we equip A and a pure subgroup G with their Z-adic topologies, then (5.1)
implies that the topology of G inherited from A is equal to its own Z-adic topology
(but the converse fails).
We will often need the concept of p-purity for a prime p. A subgroup G of A is
p-pure if
p k G D G \ p k A for every k 2 N; (5.2)
© Springer International Publishing Switzerland 2015
L. Fuchs, Abelian Groups, Springer Monographs in Mathematics,
DOI 10.1007/978-3-319-19422-6_5
149