Abelian Groups - László Fuchs [Springer]

2 Slender Groups 491
Example 2.5. The following groups are slender: Z, and more generally, every proper subgroup of
Q, and direct sums of these groups: the reduced completely decomposable groups. Also, direct
sums of any number of reduced countable torsion-free groups A i . On the other hand, Q; P are not
slender.
Monotone Subgroups Though the group P is not slender, it contains a lot of
uncountable slender subgroups, as is shown by the next theorem.
A subgroup M of P D Q n2N he ni is called a monotone subgroup (Specker [1])
if it satisfies:
(i) P n2N k ne n 2 M implies that also P n2N m ne n 2 M where m n D
max.1; jk 1 j;::::;jk n j/I
(ii) P n2N k ne n 2 M and j`nj jk n j .n 2 N/ imply P n2N `ne n 2 M.
Example 2.6.
(a) PEvery monotone subgroup contains the smallest monotone subgroup B consisting of all
n2N k ne n with jk n j bounded.
(b) The infinite sums P n2N k ne n , for which there is a c 2 N such that the coefficients satisfy
jk n jcnŠ, form a monotone subgroup.
Interestingly, monotone subgroups are slender with a single exception: the
obvious one.
Theorem 2.7 (G. Reid [1]). All monotone subgroups of the product P D Q n2N he ni
are slender, with the exception of P itself.
Proof. Let M be a monotone subgroup of P, properly contained in P, and assume
there is a homomorphism W P ! M such that e n ¤ 0 for infinitely many indices
n. We may assume that e n ¤ 0 for all n. Infinite cyclic groups are slender, so
for each k, there are but finitely many indices n for which the coordinate of e n
in he k i is non-zero. In view of this, we may even drop more summands from P
to have e n D .0;:::;0;t n;in ; t n;in C1;:::/ with t n;in ¤ 0 for a strictly increasing
sequence i 1 < < i n < of indices. Pick an x D .`1;:::;`n;:::/ 2 P n M with
1