Slender group
# 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 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 $Z^N$ 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 $Z^N.$ 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 abelian 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.
* Direct sums of slender groups are also slender.
* Subgroups of slender groups are slender.
* Every homomorphism from $Z^N$ into a slender group factors through $Z^n$ for some natural number n.
