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.

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