Slender group

In mathematics, a slender group is a torsion-free abelian group that is "small" in a sense that is made precise in the definition below.
Definition.
Let ZN denote the Baer–Specker group, that is, the group of all integer sequences, with termwise addition. For each natural number "n", let "e""n" be the sequence with "n"-th term equal to 1 and all other terms 0.
A torsion-free abelian group "G" is said to be slender if every homomorphism from ZN into "G" maps all but finitely many of the "e""n" to the identity element.
Examples.
Every free abelian group is slender.
The additive group of rational numbers Q is not slender: any mapping of the "e""n" into Q extends to a homomorphism from the free subgroup generated by the "e""n", and as Q is injective this homomorphism extends over the whole of ZN. Therefore, a slender group must be reduced.
Every countable reduced torsion-free abelian group is slender, so every proper subgroup of Q is slender.