Abelian Groups - László Fuchs [Springer]

6 Thin and Thick Groups 327
J 1 ;:::;J j ;::: are pairwise disjoint subsets in S, then[ j>m J j … S must hold for
some m 2 N. The proof of Lemma 6.4 in Chapter 2 applied to (in place of f )
convinces us that I is measurable, a contradiction.
ut
Example 6.6. To show that the last theorem may fail for measurable cardinals, let I be a
measurable index set, and A i copies of the same countable, unbounded †-cyclic p-group X, and
also C D X. Inordertodefine W A D t. Q i2I A i / ! X, choose a vector a D .:::;a i ;:::/ 2 A,
and regard the coordinates as elements of X. There are only countably many different coordinates,
so the supports of the equal ones give rise to a countable partition of I into disjoint subsets, exactly
one of which has measure 1, and the rest has measure 0. If the support of x 2 X is the one of
measure 1, then set .a/ D x. This is a well-defined homomorphism; it is not small, though i is
small for each i 2 I.
Thick Groups A p-group G is called thick (Megibben [2]) if every homomorphism
W G ! C into a †-cyclic group C is small.
Example 6.7. Torsion-complete p-groups are thick. This is a consequence of Example 6.2.
Example 6.8 (Megibben [2]). A pure dense subgroup G of a torsion-complete p-group B is thick
whenever jB=Gj @ 0 . Indeed, let W G ! C be a homomorphism, where C is †-cyclic. Then
A D Im is likewise †-cyclic. Now, extends uniquely to W B ! A, whereA denotes the
torsion-completion of A.AsIm=A is countable, by Theorem 3.5 Im is also †-cyclic. Thus is
small, and so is D G.
In the following theorem, which is a dual to Theorem 6.5, i denotes the ith
coordinate projection.
Theorem 6.9 (Rychkov–Thomé [1], Keef [6]). A p-group G is thick if and only if
the following holds:
. / for every set fA i j i 2 Ig of separable p-groups, a homomorphism W G !
˚i2I A i is small exactly if i is small for every i 2 I.
Proof. If G has property . / and the A i are cyclic p-groups, then small
implies G is thick. For the converse, assume W G ! ˚i2I A i is a non-small
homomorphism, though all of i are small. Thus for each k