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 ($\neq G$) 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 IsolatedSubgroup 2013-03-22 14:55:08 2013-03-22 14:55:08 pahio (2872) pahio (2872) 13 pahio (2872) Definition msc 20F60 msc 06A05 RankOfValuation KrullValuation rank of ordered group