# Baer-Specker group

Let $A$ be a non-empty set, and $G$ an abelian group^{}. The set $K$
of all functions from $A$ to $G$ is an abelian group, with addition^{}
defined elementwise by $(f+g)(x)=f(x)+g(x)$. The zero element^{} 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 $A=\mathbb{N}$, the set of natural numbers, and $G=\mathbb{Z}$,
$K$ as defined above is called the *Baer-Specker group*. Any
element of $K$, being a function from $\mathbb{N}$ to $\mathbb{Z}$,
can be expressed as an infinite^{} sequence^{} $({x}_{1},{x}_{2},\mathrm{\dots},{x}_{n},\mathrm{\dots})$, and the elementwise addition on $K$ can
be realized as componentwise addition on the sequences:

$$({x}_{1},{x}_{2},\mathrm{\dots},{x}_{n},\mathrm{\dots})+({y}_{1},{y}_{2},\mathrm{\dots},{y}_{n},\mathrm{\dots})=({x}_{1}+{y}_{1},{x}_{2}+{y}_{2},\mathrm{\dots},{x}_{n}+{y}_{n},\mathrm{\dots}).$$ |

An alternative
characterization^{} of the Baer-Specker group $K$ is that it can be
viewed as the countably infinite^{} direct product^{} of copies of
$\mathbb{Z}$:

$$K={\mathbb{Z}}^{\mathbb{N}}\cong {\mathbb{Z}}^{{\mathrm{\aleph}}_{0}}=\prod _{{\mathrm{\aleph}}_{0}}\mathbb{Z}.$$ |

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 |