Baer-Specker group

Let A be a non-empty set, and G an abelian groupMathworldPlanetmath. The set K of all functions from A to G is an abelian group, with additionPlanetmathPlanetmath defined elementwise by (f+g)(x)=f(x)+g(x). The zero elementMathworldPlanetmath is the function that sends all elements of A into 0 of G, and the negative of an element f is a function defined by (-f)(x)=-(f(x)).

When A=, the set of natural numbers, and G=, K as defined above is called the Baer-Specker group. Any element of K, being a function from to , can be expressed as an infiniteMathworldPlanetmath sequencePlanetmathPlanetmath (x1,x2,,xn,), and the elementwise addition on K can be realized as componentwise addition on the sequences:


An alternative characterizationMathworldPlanetmath of the Baer-Specker group K is that it can be viewed as the countably infiniteMathworldPlanetmath direct productMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of copies of :


The Baer-Specker group is an important example of a torsion-free abelian group whose rank is infinite. It is not a free abelian groupMathworldPlanetmath, but any of its countableMathworldPlanetmath subgroupMathworldPlanetmathPlanetmath is free (abelian).


  • 1 P. A. Griffith, Infinite Abelian Group Theory, The University of Chicago Press (1970)
Title Baer-Specker group
Canonical name BaerSpeckerGroup
Date of creation 2013-03-22 15:29:18
Last modified on 2013-03-22 15:29:18
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 13
Author CWoo (3771)
Entry type Definition
Classification msc 20K20
Synonym Specker group