Abelian Groups - László Fuchs [Springer]

488 13 Torsion-Free Groups of Infinite Rank
the B i are isomorphic, and the C i are non-isomorphic (Fuchs [17]). The proofs rely on the existence
of rigid systems of cardinality . A consequence is that there exist connected compact abelian
groups of cardinality 2 admitting 2 non-isomorphic closed summands which are algebraically
all isomorphic.
Honestly, for uncountable torsion-free groups, anomaly in direct decompositions is less
exciting, since their size makes it easier to use fancy constructions. However, there are certain
limitations. For instance, Kaplansky’s theorem that summands of direct sums of countable groups
are again direct sums of countable groups makes it impossible to have A D B ˚ C D ˚i A i
with indecomposable summands, where jAj D@ 1 and the A i are of finite or countable ranks. A
surprising example is a torsion-free group which is both the countable direct sum and the countable
direct product of copies of the very same group; see Eda’s Proposition 4.9.
Göbel–Ziegler [1] prove that, for every infinite cardinal , there exist groups of cardinality
that decompose into the direct sum of non-zero summands for each 1
is relevant.
(c) Part (a) fails to hold if the components are not all of finite rank.
(3) (Corner) There are countable torsion-free groups A; B; C such that A Š B ˚ C;
B Š A ˚ C; but A and B are not isomorphic. [Hint: choose m D 2 in Theorem
1.6.]
(4) (Corner) There exists a countable torsion-free group A such that A is isomorphic
to the direct sum of any finite number of, but not to the direct sum of infinitely
many copies of itself.
(5) If A is superdecomposable, then so are the groups A ./ for every cardinal .
[Hint: direct sums of groups in Theorem 1.5; prove this for a finite sum, a
countable sum, and apply Theorem 2.5 in Chapter 2.]