Abelian Groups - László Fuchs [Springer]

4 Separable Groups 501
Exercises
(1) An @ 1 -free group is slender if and only if it has no subgroup Š P.
(2) (Nunke) Equip P with the product topology where the components he n i carry
the discrete topology. Then: (a) all the endomorphisms of P are continuous; (b)
the topology of P is independent of the way P is represented as a direct product
of infinite cyclic groups.
(3) If G is a dense subgroup of P,thenP=G is cotorsion.
(4) A subgroup of P that is closed in the product topology of P is a product of
infinite cyclic groups. [Hint: Proof of Proposition 3.6.]
(5) Suppose X is a product in P. ThenP=X is a product of finite and infinite cyclic
groups.
(6) Show that J p does not contain any subgroup Š P.
(7) Let A be a torsion-free group that contains a slender subgroup G such that A=G
is a reduced torsion group. Show that A is slender. [Hint: Im is reduced
torsion cotorsion (Proposition 3.6), so bounded; W P ! A, W A ! A=G.]
(8) A countable group is cotorsion-free if and only if it is slender.
4 Separable Groups
Our next topic is a class of groups that is more general than completely decomposable
groups, but behave “locally” as if they were completely decomposable.
A torsion-free group A is called separable (Baer [6]) if every finite subset of
elements of A is contained in a completely decomposable summand of A. Clearly,
this summand may then be assumed to be of finite rank. Recall: if A has a rank 1
summand of type t,thenwesayt is an extractable type in A.
Example 4.1. The proof of Theorem 8.2 in Chapter 3 shows that the Baer-Specker group Z @0 is
separable, but not completely decomposable. The same holds for the generalized Baer-Specker
group Z for any infinite cardinal .
Properties of Separability Some key properties of separability are listed below.
(A) A group is separable if and only if its reduced part is separable.
(B) Direct sums of separable groups are separable (but their direct products are
not necessarily).
(C) If A is separable, then for every type t, A.t/ and A .t/ are pure, separable
subgroups of A.
(D) If A is separable, and t is any type, then A.t/=A .t/ is separable. This will be
a consequence of the following lemma.
Lemma 4.2.
(i) Fully invariant subgroups of a separable group are separable.
(ii) Factor groups of a separable group modulo fully invariant pure subgroups are
separable.