Slender group
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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.
[edit] Definition
Let ZN denote the Baer-Specker group, that is, the group of all integer sequences, with termwise addition. For each n in N, let en 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 ZN into G maps all but finitely many of the en to the identity element.
[edit] Examples
Every free abelian group is slender.
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.
[edit] References
- Fuchs, László (1973), Infinite abelian groups. Vol. II, Boston, MA: Academic Press, MR0349869, especially chapter XIII.
- R. J. Nunke, Slender groups, Bulletin of the American Mathematical Society, vol. 67 (1961), 274-275.
- Saharon Shelah and Oren Kolman, Infinitary axiomatizability of slender and cotorsion-free groups, Bull. Belg. Math. Soc. Simon Stevin, vol. 7 (2000), 623-629.
