Abelian Groups - László Fuchs [Springer]

6 Powers ofZ of Measurable Cardinalities 513
Mishina [1] proves that every slender summand of a vector group is a finite direct sum of its
summands. As pointed out above, separable vector groups have been characterized by Mishina [2]
and Król [1] independently. (I wish to apologize for quoting their result incorrectly in [IAG].) See
also Król–Sa¸siada [1]. Albrecht–Hill [2] provide a better proof.
Huber [1] investigates homomorphisms W R I ! R J for rational groups R and non-measurable
index sets I; J. The kernel and cokernels are characterized: they are of the form Ker D Hom.A; R/
and Coker D R K ˚ Ext.A; R/ for some group A and index set K. Ivanov [7] shows that
Kaplansky’s test problems have positive solutions for non-measurable vector groups.
Exercises
(1) For non-measurable I, derive Lemma 5.4(b) directly from Corollary 2.10,
without referring to (a).
(2) (Mishina, Łoś) A vector group over an infinite index set is completely decomposable
if and only if almost all components are Š Q.
(3) A reduced vector group contains no cotorsion subgroups ¤ 0.
(4) Let R be a rational group of idempotent type, and a non-measurable cardinal.
There exists a group A of cardinality such that Hom.A; R/ Š A.
(5) (Beaumont) Let V D Q i2I R i be an elementary vector group of type t with
infinite I. Show that V contains elements of any type s satisfying t 0 s t.
(Here, as above, t 0 D t W t.)
(6) Prove that .Z @ 0
/ n Š Z @ 0
for each n 2 N,but.Z @ 0
/ .@ 0/ 6Š Z @ 0
.
(7) (Ivanov) Two non-measurable vector groups are isomorphic if each is isomorphic
to a summand of the other.
6 Powers of Z of Measurable Cardinalities
The discussion of vector groups above suggests that there is a fundamental
difference between the behavior of vector groups of measurable and non-measurable
cardinalities. Our study would be incomplete without mentioning some results that
are critical in understanding why non-measurability was so relevant an assumption
in the above theorems. Already the measurable powers Z I display phenomena whose
understanding requires heavy set-theoretical machinery. To study Z I fully in detail
would take us too far afield, so we restrict ourselves to pointing out some typical
results.
In this section, we are working in a model of set theory in which measurable
cardinals exist. We will use the symbol for the smallest measurable cardinal,
and the notation P for the th power Z of Z which will also be written more
explicitly as
P D Y i< he ii where he i iŠZ