// Workers AI · dad joke modeWhat did the slender group say? We're a thin crowd.

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 $\mathbb {Z} ^{\mathbb {N} }$ 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 $n$.

References

• Fuchs, László (1973). Infinite abelian groups. Vol. II. Pure and Applied Mathematics. Vol. 36. Boston, MA: Academic Press. Chapter XIII. MR 0349869. Zbl 0257.20035..
• Griffith, Phillip A. (1970). Infinite Abelian group theory. Chicago Lectures in Mathematics. University of Chicago Press. pp. 111–112. ISBN 0-226-30870-7. Zbl 0204.35001.
• Nunke, R. J. (1961). "Slender groups". Bulletin of the American Mathematical Society. 67 (3): 274–275. doi:10.1090/S0002-9904-1961-10582-X. Zbl 0099.01301.
• Shelah, Saharon; Kolman, Oren (2000). "Infinitary axiomatizability of slender and cotorsion-free groups". Bulletin of the Belgian Mathematical Society. 7: 623–629. MR 1806941. Zbl 0974.03036.