A torsion-free abelian group G is said to be slender if every homomorphism from ZN into G maps all but finitely many of the en to the identity element.
Every free abelian group is slender.
The additive group of rational numbers Q is not slender: any mapping of the en into Q extends to a homomorphism from the free subgroup generated by the en, and as Q is injective this homomorphism extends over the whole of ZN. Therefore, a slender group must be reduced.
Every countable reduced torsion-free abelian group is slender, so every proper subgroup of 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 p-adic integers for any p.
- Direct sums of slender groups are also slender.
- Subgroups of slender groups are slender.
- Every homomorphism from ZN into a slender group factors through Zn for some natural number n.
- Fuchs, László (1973). Infinite abelian groups. Vol. II. Boston, MA: Academic Press. MR 0349869. See especially chapter XIII.
- Phillip A. Griffith (1970). Infinite Abelian group theory. Chicago Lectures in Mathematics. University of Chicago Press. pp. 111–112. ISBN 0-226-30870-7.
- R. J. Nunke (1961). "Slender groups". Bulletin of the American Mathematical Society 67 (3): 274–275. doi:10.1090/S0002-9904-1961-10582-X.
- Saharon Shelah; Oren Kolman (2000). "Infinitary axiomatizability of slender and cotorsion-free groups". Bulletin of the Belgian Mathematical Society 7: 623–629.
|This abstract algebra-related article is a stub. You can help Wikipedia by expanding it.|