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 {\displaystyle \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 {\displaystyle n}, let e n {\displaystyle e_{n}} be the sequence with n {\displaystyle n}-th term equal to 1 and all other terms 0.

A torsion-free abelian group G {\displaystyle G} is said to be slender if every homomorphism from Z N {\displaystyle \mathbb {Z} ^{\mathbb {N} }} into G {\displaystyle G} maps all but finitely many of the e n {\displaystyle e_{n}} to the identity element.

Examples

Every free abelian group is slender.

The additive group of rational numbers Q {\displaystyle \mathbb {Q} } is not slender: any mapping of the e n {\displaystyle e_{n}} into Q {\displaystyle \mathbb {Q} } extends to a homomorphism from the free subgroup generated by the e n {\displaystyle e_{n}}, and as Q {\displaystyle \mathbb {Q} } is injective this homomorphism extends over the whole of Z N {\displaystyle \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 Q {\displaystyle \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 {\displaystyle p}-adic integers for any p {\displaystyle p}.
• Direct sums of slender groups are also slender.
• Subgroups of slender groups are slender.
• Every homomorphism from Z N {\displaystyle \mathbb {Z} ^{\mathbb {N} }} into a slender group factors through Z n {\displaystyle \mathbb {Z} ^{n}} for some natural number n {\displaystyle n}.

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