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# isolated subgroup

Let $G$ be a multiplicative ordered group and $F$ its subgroup. We call this subgroup isolated 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 rank of $G$.

###### Theorem.

Let $G$ be an abelian ordered group with order at least 2. The rank 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).

Defines:

rank of ordered group

Related:

RankOfValuation, KrullValuation

Type of Math Object:

Definition

Major Section:

Reference

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## Mathematics Subject Classification

20F60*no label found*06A05

*no label found*

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