Abelian Groups - László Fuchs [Springer]

3 Torsion-Complete Groups 313
In particular, a separable p-group A with basic subgroup B is isomorphic to a
pure .dense/ subgroup of B containing B.
Proof. It remains to check the purity of Im in B. By Sect. 6(F) in Chapter 5, B is
a basic subgroup in A, soA=B is a divisible group. Manifestly, B is basic in B,
whence the purity of A in B is immediate.
ut
It is appropriate to regard the torsion-complete group B obtained from the p-
group A in the preceding theorem as the torsion-completion of A. It is unique up to
isomorphism.
As a consequence, a separable p-group may be thought of (and treated as) a
pure and dense subgroup of a torsion-complete group. This point of view gives us
considerably more leverage than regarding it as a subgroup of a divisible group. To
illustrate this, we interrupt momentarily the discussion of our main topic, and prove
the following slight generalization of Prüfer’s Theorem 5.3 in Chapter 3.
Theorem 3.5. Let A be a separable p-group, and B a pure †-cyclic subgroup of A.
If A=B is countable, then A is †-cyclic.
Proof. As B can be expanded to be a basic subgroup, we may assume B is a
basic subgroup of A. A is treated as a pure subgroup of B containing B. Let
a 1 ;:::;a m ;::: be a countable set that, together with B, generates A. We can write
a m D .b m1 ;:::;b mn ;:::/ 2 B with b mn 2 B n . Each B n is a direct sum of cyclic
groups of order p n , so there is a decomposition B n D B 0 n ˚ B00 n such that b mn 2 B 0 n
for all m, andB 0 n is countable. Setting B0 D˚n B 0 n and B00 D˚n B 00
n , we obtain
B D B 0˚B00 and A D A 0˚B00 where A 0 DhB 0 ; a 1 ;:::;a m ;:::i.HereA 0 is countable
and separable, so by Theorem 5.3 in Chapter 3 it is †-cyclic.
ut
Theorems on Torsion-Complete Groups We return to torsion-complete
groups, and continue with various algebraic characterizations. The main algebraic
features are encapsulated in the following theorem.
Theorem 3.6. For a reduced p-group A, the following conditions are equivalent.
(i) A is torsion-complete;
(ii) A is the torsion subgroup of a reduced algebraically compact group;
(iii) A is pure-injective in the category of p-groups .i.e., it has the injective property
relative to pure-exact sequences of p-groups/;
(iv) A is a direct summand in every p-group in which it is contained as a pure
subgroup.
Proof.
(i) ) (ii) This implication is obvious, since B is by definition the torsion subgroup
in the algebraically compact (= complete) group QB.
(ii) ) (iii) Next, suppose (ii), i.e. A is the torsion part of an algebraically compact
group C. Let0 ! G ˛!H ˇ!K ! 0 be a pure-exact sequence of p-groups,
and W G ! A a homomorphism. If is viewed as a map G ! C, thenby