Baer–Specker group

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In mathematics, in the field of group theory, the Baer–Specker group, or Specker group, named after Reinhold Baer and Ernst Specker, is an example of an infinite Abelian group which is a building block in the structure theory of such groups.

Definition[edit]

The Baer–Specker group is the group B = ZN of all integer sequences with componentwise addition, that is, the direct product of countably many copies of Z.

Properties[edit]

Reinhold Baer proved in 1937 that this group is not free abelian; Specker proved in 1950 that every countable subgroup of B is free abelian.

The group of homomorphisms from the Baer–Specker group to a free abelian group of finite rank is a free abelian group of countable rank. This provides another proof that the group is not free.[1]

See also[edit]

Notes[edit]

  1. ^ Blass & Göbel (1994) attribute this result to Specker (1950). They write it in the form P^*\cong S where P denotes the Baer-Specker group, the star operator gives the dual group of homomorphisms to \mathbb{Z}, and S is the free abelian group of countable rank. They continue, "It follows that P has no direct summand isomorphic to S", from which an immediate consequence is that P is not free abelian.

References[edit]

  • Baer, Reinhold (1937), "Abelian groups without elements of finite order", Duke Mathematical Journal 3 (1): 68–122, doi:10.1215/S0012-7094-37-00308-9, MR 1545974 .
  • Blass, Andreas; Göbel, Rüdiger (1996), "Subgroups of the Baer-Specker group with few endomorphisms but large dual", Fundamenta Mathematicae 149 (1): 19–29, arXiv:math/9405206, MR 1372355 .
  • Specker, Ernst (1950), "Additive Gruppen von Folgen ganzer Zahlen", Portugaliae Math. 9: 131–140, MR 0039719 .
  • Griffith, Phillip A. (1970), Infinite Abelian group theory, Chicago Lectures in Mathematics, University of Chicago Press, pp. 1, 111–112, ISBN 0-226-30870-7 .

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