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Baer-Specker group
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(Definition)
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Let  be a non-empty set, and  an abelian group. The set  of all functions from  to  is an abelian group, with addition defined elementwise by
 . The zero element is the function that sends all elements of  into 0 of  , and the negative of an element  is a function defined by
 .
When
 , the set of natural numbers, and
 ,  as defined above is called the Baer-Specker group. Any element of  , being a function from
 to
 , can be expressed as an infinite sequence
 , and the elementwise addition on  can be realized as componentwise addition on the sequences:
An alternative characterization of the Baer-Specker group  is that it can be viewed as the countably infinite direct product 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 group, but any of its countable subgroup is free (abelian).
- 1
- P. A. Griffith, Infinite Abelian Group Theory, The University of Chicago Press (1970)
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"Baer-Specker group" is owned by CWoo.
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(view preamble)
| Other names: |
Specker group |
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Cross-references: abelian, subgroup, countable, free abelian group, rank, torsion-free, direct product, countably infinite, characterization, sequence, infinite, natural numbers, negative, zero element, addition, functions, abelian group
This is version 10 of Baer-Specker group, born on 2005-08-24, modified 2005-10-24.
Object id is 7344, canonical name is BaerSpeckerGroup.
Accessed 2387 times total.
Classification:
| AMS MSC: | 20K20 (Group theory and generalizations :: Abelian groups :: Torsion-free groups, infinite rank) |
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Pending Errata and Addenda
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