Slender group

Short description: Mathematical term

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

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

References

• Fuchs, László (1973). Infinite abelian groups. Vol. II. Pure and Applied Mathematics. 36. Boston, MA: Academic Press. Chapter XIII. .
• Griffith, Phillip A. (1970). Infinite Abelian group theory. Chicago Lectures in Mathematics. University of Chicago Press. pp. 111–112. ISBN 0-226-30870-7. 
• Nunke, R. J. (1961). "Slender groups". Bulletin of the American Mathematical Society 67 (3): 274–275. doi:10.1090/S0002-9904-1961-10582-X. 
• Shelah, Saharon; Kolman, Oren (2000). "Infinitary axiomatizability of slender and cotorsion-free groups". Bulletin of the Belgian Mathematical Society 7: 623–629. http://projecteuclid.org/euclid.bbms/1103055621.