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 Z denote the Baer–Specker group, that is, the group of all integer sequences, with termwise addition. For each natural number "n", let "e" 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 Z into "G" maps all but finitely many of the "e" 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" into Q extends to a homomorphism from the free subgroup generated by the "e", and as Q is injective this homomorphism extends over the whole of Z. Therefore, a slender group must be reduced. Every countable reduced torsion-free abelian group is slender, so every proper subgroup of Q is slender.