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.

Definition[edit]

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.

Examples[edit]

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.

Properties[edit]

  • 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.

References[edit]

  • 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.