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^N denote the Baer–Specker group, that is, the group of all integer sequences, with termwise addition. For each "n" in 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 Z^N into "G" maps all but finitely many of the "e"_"n" to the identity element.

Examples

Every free abelian group is slender.

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 Z^N. Therefore, a slender group must be reduced.

Every countable reduced torsion-free abelian group is slender, so every proper subgroup of Q is slender.

Properties

* A torsion-free group is slender if and only if it is reduced and contains no copy of the Baer–Specker group and no copy of the p-adic integers for any "p".

References

* | year=1973 See especially chapter XIII.
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