Abelian Groups - László Fuchs [Springer]

164 5 Purity and Basic Subgroups
Theorem 4.3 (Maranda [1]). A group is pure-projective if and only if it is
†-cyclic.
Proof. Theorem 3.2 implies that †-cyclic groups are pure-projective.
Conversely, assume that A is a pure-projective group. By Lemma 4.2, there exists
a pure-exact sequence (5.5) with †-cyclic P. By pure-projectivity, there is a map
˛ W A ! P such that ˛ D 1 A . This means, A is isomorphic to a summand of P,and
hence it is †-cyclic.
ut
Pure-Injective Groups Turning to the dual concept, a group H is defined to be
pure-injective if it has the injective property relative to all pure-exact sequences. In
other words, any diagram
0 −−−−→ A
⏐
φ↓
H
α
−−−−→
ψ
B
β
−−−−→ C −−−−→ 0
with pure-exact row can be completed with a map W B ! H such that ˛ D .
As we shall see, the theory of pure-injective groups can be incorporated in the theory
of algebraically compact groups, so here we restrict ourselves to a few elementary
results. For more information, we refer to Sect. 4 in Chapter 6.
Example 4.4. Injective groups are trivially pure-injective, and so are all the cocyclic groups (cf.
Theorem 3.2).
Example 4.5. The group J p is pure-injective. This will follow from Theorem 1.2 in Chapter 6 as
J p is a compact group.
The next lemma provides pure-injective resolutions (so there are enough pureinjectives).
Lemma 4.6 (Łoś [1]). Every group can be embedded as a pure subgroup in a direct
product of cocyclic groups.
Proof. Let fH i .i 2 I/g be the set of all cocyclic factor groups of the group A,andset
H D Q i2I H i. The canonical maps i W A ! H i induce a homomorphism W A ! H
which must be an embedding, since every non-zero a 2 A is excluded from the
kernel of some i (see Proposition 5.5 in Chapter 4). To verify the purity of Im
in H, we show that if a 2 A is such that a … p n A,thenalsoa … p n H.LetC be
a subgroup of A maximal with respect to the properties p n A C and a … C. Then
by Proposition 5.5 in Chapter 4 A=C is cocyclic, and since it is bounded, it must be
cyclic of order p k for some k n. Thus A=C D H i for some i,where i a is of height
k 1,sothata 2 p n H is impossible. ut