Abelian Groups - László Fuchs [Springer]

4 Indecomposable Groups 435
Lemma 4.7 (Fuchs [12]). Let fE 0 ; E i .i 2 I/g be a rigid system of torsion-free
groups, and p i .i 2 I/ primes such that there are elements u i 2 E 0 and v i 2 E i not
divisible by p i . Then the group
A DhE 0 ˚ .˚i2I E i /I pi
1 .u i C v i / 8 i 2 Ii
is indecomposable. The same holds if pi 1 is replaced by pi 1 or by ni
1 for integers
n i whose prime divisors divide neither u i nor v i .
Proof. The proof of indecomposability runs as before. Rigity ensures the full
invariance of the subgroups E 0 ; E i in A. Therefore, if A D B ˚ C, thenE i D
.B \ E i / ˚ .C \ E i / for i 2 I [f0g. Hence the indecomposability of the E i implies
that each of E i is contained entirely either in B or in C. Assume E 0 B and E j C
for some j 2 I, and write pj
1 .u j C v j / D b C c .b 2 B; c 2 C/. Hence u j D p j b and
v j D p j c, which is impossible in E j , and likewise in A. It follows that all of E 0 ; E i
belong to the same summand B or C,andA is indecomposable.
ut
The Blowing Up Lemma We keep in mind that our ultimate goal in this section
is to establish the existence of arbitrarily large indecomposable groups. In order to
get ready for the proof, we will need a powerful result on blowing up a small system
of indecomposable groups to extremely large systems. First, a definition.
Let S denote a Z-algebra (in our application, S D Z). A fully rigid system
fG X j X Ig of groups for S is a set of groups G X , indexed by the subsets X of a set
I, such that
(i) G X G Y whenever X Y, and
(ii) Hom.G X ; G Y / Š S or 0 according as X Y or not.
(If S is a subring of Q, then a fully rigid system for S is a rigid system in the sense
above.) We quote the relevant result without proof.
Lemma 4.8 (Corner [8]). Suppose there exists a fully rigid system fN X j X Jg
of groups for the Z-algebra S over a set J of at least six elements. If I is any infinite
set of cardinality jN J j, then there exists a fully rigid system fM X j X Ig for S
such that
jM X jDjIj for all X I:
Evidently, all the groups in a fully rigid system are indecomposable if 0 and 1
are the only idempotents in S.
Arbitrarily Large Indecomposable Groups By virtue of the preceding lemma,
in order to prove the existence of arbitrarily large indecomposable groups, it only
remains to find a fully rigid system for Z over a set with at least six elements. This
is what we are going to do in the next proof.
Theorem 4.9. (i) (Shelah [1]) For any cardinal , there exist indecomposable
torsion-free groups of cardinality .