# isolated subgroup

Let $G$ be a ordered group and $F$ its subgroup^{}. We call this subgroup if every element $f$ of $F$ and every element $g$ of $G$ satisfy

$$f\leqq g\leqq 1\Rightarrow g\in F.$$ |

If an ordered group $G$ has only a finite number of isolated subgroups, then the number of proper ($\ne G$) isolated subgroups of $G$ is the of $G$.

###### Theorem.

Let $G$ be an abelian^{} ordered group with order (http://planetmath.org/OrderGroup) at least 2. The of $G$ equals one iff there is an order-preserving isomorphism^{} from $G$ onto some subgroup of the multiplicative group^{} of real numbers.

## References

- 1 M. Larsen & P. McCarthy: Multiplicative theory of ideals. Academic Press. New York (1971).

Title | isolated subgroup |
---|---|

Canonical name | IsolatedSubgroup |

Date of creation | 2013-03-22 14:55:08 |

Last modified on | 2013-03-22 14:55:08 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 13 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 20F60 |

Classification | msc 06A05 |

Related topic | RankOfValuation |

Related topic | KrullValuation |

Defines | rank of ordered group |