Abelian Groups - László Fuchs [Springer]

514 13 Torsion-Free Groups of Infinite Rank
with projection maps i W P !he i i.Asusual,wesetS D˚i< he i i.Anelement
of P will be written in the form x D P i< n ie i with n i 2 Z.
Let U be a (non-principal) @ 1 -complete ultrafilter on the power set P.I/ of an
index set I of cardinality . The homomorphism U W P ! Z is defined as follows.
For x 2 P ,letI n Dfi 2 I j i .x/ D ne i g for n 2 Z, sothatfI n j n 2 Zg is a
partition of I. Ifm is the unique integer such that I m 2 U, thenweset U .x/ D m: It
is straightforward to check that U is a genuine homomorphism.
It is known that if the cardinal is measurable, then there are at least 2 -
complete ultrafilters on .
Uniqueness of in P If we consider powers Z , then probably one of the
first questions that comes to mind is to what extent is determined. We verify the
uniqueness of this cardinality in a more general setting.
Theorem 6.1 (Eda [1]). Let G ¤ 0 denote a slender group. G I Š G J holds for
infinite sets I; J if and only if jIj DjJj.
Proof. To verify the “only if” part, let W G I ! G J be an isomorphism, and j W
G J ! G the jth projection map. Pick a 0 ¤ g 2 G, and for each i 2 I, letg i 2 G I
be the vector with 0 coordinates everywhere except with g as its ith coordinate. As
G is slender, for any fixed j 2 J, thesetofi 2 I satisfying j ..g i // ¤ 0 must be
finite. Consequently, jIj jJj, and by symmetry, we obtain the desired equality. ut
Odd Decompositions of Z In order to illustrate how drastically the measurable
case differs from the non-measurable situation, we exhibit examples showing that
for a measurable cardinal , P admits non-trivial direct decompositions where S
is contained in one of the summands.
Example 6.2 (O’Neill [3]).
(a) Let be a measurable cardinal, and let U denote a non-principal @ 1 -complete ultrafilter on
the power set P./. ThenP D A ˚hei, whereA Dfx 2 P j supp x … Ug, ande 2 P is
the vector whose ith coordinate is e i for each i