Slender group

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 Z^N denote the Baer–Specker group, that is, the group of all integer sequences, with termwise addition. For each n in 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 Z^N into G maps all but finitely many of the e_n to the identity element.

ExamplesEvery countable reduced torsion-free abelian group is slender, so every proper subgroup of Q is slender.

Every free abelian group is slender.

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

Every countable reduced torsion-free abelian group is slender, so every proper subgroup of 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 Z^N into a slender group factors through Zⁿ 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. 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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