Abelian Groups - László Fuchs [Springer]

492 13 Torsion-Free Groups of Infinite Rank
Theorem 2.8 (Łoś). Let G be a slender group, and W P ! G. Then
(i) A i D 0 for almost all i,
(ii) if is not measurable and S D 0,then D 0.
Proof.
(i) Suppose, by way of contradiction, that A i ¤ 0 holds for infinitely many
indices, say, for i D i 1 ;:::;i n ;:::. Then choosing e n 2 A in with e n ¤ 0, the
restriction of to P 0 D Q 1
nD1 he ni. P / would be a forbidden homomorphism
into a slender group.
(ii) This is also an indirect proof, but more sophisticated. Suppose that S D 0,
but there exists an a 2 P with a ¤ 0. We introduce a G-valued “measure”
on the subsets of I, by using the selected element a. For a subset J of I, we
set .J/ D a J ,wherea J has the same coordinates as a for indices j 2 J, and
0 coordinates elsewhere. It is clear that for pairwise disjoint subsets J; K of I,
we have .J [ K/ D .a J[K / D .a J C a K / D .J/ C .K/, sothat is
additive. In order to convince ourselves that it is moreover countably additive,
let J 1 ;:::;J n ;:::be a countable set of pairwise disjoint subsets of I, andJ 0 the
complement of their union J in I. ThenP D Q 1
kD0 ha J k
i is a subgroup of P .
Applying the definition of slenderness to P W P ! G, we conclude that
almost all of a Jk D .J k / D 0. This shows that is countably additive. An
appeal to Lemma 6.4 in Chapter 2 shows that I is measurable, completing the
proof.
ut
If P is equipped with the product topology (keeping the components discrete),
then condition (i) in Theorem 2.8 amounts to that every homomorphism of P into
a (discrete) slender group is continuous. We should point out that (ii) is the best
possible result in the sense that for measurable cardinals , S D 0 no longer
implies D 0. This is illustrated by the following example, generalizing Łoś’
original example.
Example 2.9. For a measurable index set I, there are non-trivial homomorphisms of P D Z I
into any torsion-free group G ¤ 0 such that S D 0 where S D Z .I/ . To prove this, we choose
any 0 ¤ g 2 G. Let be a f0; 1g-valued measure on I, andfora D .:::;k i ;:::/ 2 P define
pairwise disjoint subsets X n .a/ I .n 2 Z/ via X n .a/ Dfi 2 I j k i D ng. Then the union of these
subsets is I, hence exactly one of them, say, X m .a/ has measure 1, and the rest have measure 0. We
then set a D mg, thus P ¤ 0: From the properties of it is readily checked that preserves
addition, and S D 0.
At this point let us pause and list a couple of corollaries to the last theorem.
Corollary 2.10. Let G be a slender group, and A i .i 2 I/ torsion-free groups. If the
index set I is non-measurable, then there is a natural isomorphism
Y
Hom A i; G Š˚i2I Hom.A i ; G/:
i2I