Abelian Groups - László Fuchs [Springer]

2 Slender Groups 489
(6) A torsion-free group of infinite rank is a subdirect product of copies of Q.
(7) (Reid) Every torsion-free group of infinite rank is a sum of two free groups.
2 Slender Groups
A remarkable class of torsion-free groups was discovered by J. Łoś in the 1950s:
the class of slender groups. Without further ado, we embark on their very attractive
theory.
Let P denote the direct product of a countable set of infinite cyclic groups, and S
its subgroup that is the direct sum of the same groups:
1Y
P D he n i;
nD1
1M
S D he n i;
nD1
where he n iŠZ:
The elements of P can be written as formal infinite sums: x D P 1
nD1 k ne n ,oras
infinite vectors, like x D .k 1 e 1 ;:::;k n e n ;:::/, or else simply as x D .k 1 ;:::;k n ;:::/,
where k n 2 Z.
Slenderness A group G is called slender if, for every homomorphism
W P ! G, wehavee n D 0 for almost all indices n. From this definition it
is evident that the following hold true:
(a) Subgroups of slender groups are slender.
(b) The group P is not slender.
(c) No injective group ¤ 0 is slender, so slender groups are reduced.
(d) The group J p of p-adic integers, and more generally, an algebraically compact
group ¤ 0, is not slender. In fact, the free group S can be mapped into J p by a
homomorphism such that e n ¤ 0 for all n. SinceS is pure in P, andJ p is
algebraically compact, extends to a map W P ! J p with e n ¤ 0 for all n.
(e) Slender groups are torsion-free. The argument in (d) works for Z.p/ (for any
prime p), since it is algebraically compact. A slender group cannot contain a
non-slender Z.p/, so it must be torsion-free.
The following lemma is a consequence of our remarks.
Lemma 2.1. If W P ! G with G slender, then S D 0 implies D 0.
Homomorphic image of P in a slender group is a finitely generated free group.
Proof. If is as stated, then Im is an epic image of the algebraically compact
group P=S, so it is a cotorsion group. A torsion-free cotorsion group is algebraically
compact. No such group can be a subgroup of G, hence Im D 0. The second claim
is a simple corollary to the first, using the definition of slenderness.
ut