Baer-Specker group
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 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).
References
- 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 |