Baer–Specker group

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 = Z^N of all integer sequences with componentwise addition, that is, the direct product of countably infinitely many copies of Z. It can equivalently be described as the additive group of formal power series with integer coefficients.

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]

Notes

^ Blass & Göbel (1996) 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, hdl:10338.dmlcz/100591, 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, Bibcode:1994math......5206B, doi:10.4064/fm-149-1-19-29, MR 1372355, S2CID 18281146.
Specker, Ernst (1950), "Additive Gruppen von Folgen ganzer Zahlen", Portugaliae Mathematica, 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.
Cornelius, E. F., Jr. (2009), "Endomorphisms and product bases of the Baer-Specker group", Int'l J Math and Math Sciences, 2009, article 396475, https://www.hindawi.com/journals/ijmms/

External links

Stefan Schröer, Baer's Result: The Infinite Product of the Integers Has No Basis

[1] Blass & Göbel (1996) 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.

[1]