Baer–Specker group

Definition

The Baer–Specker group is the group B=\mathbb{Z}^\mathbb{N} of all integer sequences with componentwise addition, that is, the direct product of countably infinitely many copies of \mathbb Z. n be the sequence with n-th term equal to 1 and all other terms 0.-- 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.

Notes

References

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• 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/

References

1. {{harvtxt. Blass. Göbel. 1996 attribute this result to {{harvtxt. Specker. 1950. They write it in the form $P^*\backslash cong\; S$ where $P$ denotes the Baer-Specker group, the star operator gives the dual group of homomorphisms to $\backslash mathbb\{Z\}$, 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.