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.

Let $\mathbb {Z} ^{\mathbb {N} }$ 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 $\mathbb {Z} ^{\mathbb {N} }$ 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 $\mathbb {Q}$ is not slender: any mapping of the $e_{n}$ into $\mathbb {Q}$ extends to a homomorphism from the free subgroup generated by the $e_{n}$ , and as $\mathbb {Q}$ is injective this homomorphism extends over the whole of $\mathbb {Z} ^{\mathbb {N} }$ . Therefore, a slender group must be reduced.

Every countable reduced torsion-free abelian group is slender, so every proper subgroup of $\mathbb {Q}$ 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 $\mathbb {Z} ^{\mathbb {N} }$ into a slender group factors through $\mathbb {Z} ^{n}$ for some natural number $n$ .

Examples

Properties

References

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