Abelian Groups - László Fuchs [Springer]

484 13 Torsion-Free Groups of Infinite Rank
each subset gives rise to a direct decomposition into indecomposable summands
whose ranks are exactly the elements in the selected subset. (We could have admitted
repeated use of the same rank, but there was no need for it.)
The next theorem shows that a countable group may have continuously many
non-isomorphic indecomposable summands even if it is a direct sum of only two
indecomposable groups.
Theorem 1.4 (Fuchs [17]). There exists a torsion-free group A of rank @ 0 such that
A D B j ˚ C j with B j Š C j
holds for continuously many, pairwise non-isomorphic indecomposable groups B j
of countable rank.
Proof. Let P i ; Q i .i 2 N/ be pairwise disjoint, infinite sets of primes and p; q; r
distinct odd primes not in their union. With independent elements a i ; b i ; c i ; d i .i 2 N/
we define
B DhPi 1 a i ; Qi
1 b i ; p 1 .a i C a iC1 /; q 1 .b i C b iC1 /; r 1 .a i C b i / 8 i 2 Ni;
C DhPi 1 c i ; Qi
1 d i ; p 1 .c i C c iC1 /; q 1 .d i C d iC1 /; r 1 .c i C d i / 8 i 2 Ni:
Then B and C are isomorphic indecomposable groups, and set A D B ˚ C, For each
i choose an integer k i (to be specified later), and let
u i D a i ; v i D k i b i C .k 2 i
1/d i ; x i D k i a i C c i ; y i D b i C k i d i
for all i 2 N. The aim is to choose the integers k i such that A D U ˚ X where
U DhPi 1 u i ; Qi
1 v i ; p 1 .u i C u iC1 /; q 1 .v i C v iC1 /; r 1 .u i C k i v i / 8 i 2 Ni;
X DhPi 1 x i ; Qi
1 y i ; p 1 .x i C x iC1 /; q 1 .y i C y iC1 /; r 1 .x i C k i y i / 8 i 2 Ni:
Investigating divisibility by p; q; and r, our usual technique shows that for A D
U ˚ X it is necessary and sufficient that the k i be subject to the conditions:
k i k iC1 mod pq and k 2 i 1 mod r for all i 2 N: (13.1)
Pick an integer ` such that ` 1 mod pq and ` 1 mod r. If, for each i, we
choose k i D 1 or k i D `, then the sequence of such k i will satisfy (13.1), so we get
a decomposition A D U ˚ X with indecomposable U Š X.
We fix k 1 D 1, and show that if the sequence k 2 ;:::;k i ;::: differs from the
sequence k2 0 ;:::;k0 i ;::: (i.e., for at least one i, k i ¤ ki 0 ), then the corresponding