Hellenica World

# .

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

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

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]

Slender group

Notes

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

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.