Slender group

Let ${\displaystyle \mathbb {Z} ^{\mathbb {N} }}$ denote the Baer–Specker group, that is, the group of all integer sequences, with termwise addition. For each natural number ${\displaystyle n}$, let ${\displaystyle e_{n}}$ be the sequence with ${\displaystyle n}$-th term equal to 1 and all other terms 0.

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

Every free abelian group is slender.

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

Every countable reduced torsion-free abelian group is slender, so every proper subgroup of ${\displaystyle \mathbb {Q} }$ is slender.

• 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 ${\displaystyle p}$-adic integers for any ${\displaystyle p}$.
• Direct sums of slender groups are also slender.
• Subgroups of slender groups are slender.
• Every homomorphism from ${\displaystyle \mathbb {Z} ^{\mathbb {N} }}$ into a slender group factors through ${\displaystyle \mathbb {Z} ^{n}}$ for some natural number ${\displaystyle n}$.

• 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.

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