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isolated subgroup

(Definition)

Let be a multiplicative ordered group and its subgroup. We call this subgroup isolated if every element of and every element of satisfy

$\displaystyle f \leqq g \leqq 1 \,\,\,\Rightarrow \,\, g\in F.$

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

Theorem 1 Let

be an abelian ordered group with order at least 2. The rank of

equals one iff there is an order-preserving isomorphism from

Bibliography

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

"isolated subgroup" is owned by pahio.

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Also defines:

rank of ordered group

This object's parent.

Attachments:: characterization of ordered groups of rank one (Theorem) by rspuzio
proof of embedding theorem for ordered abelian groups of rank one (Proof) by rspuzio

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Cross-references: real numbers, multiplicative group, onto, isomorphism, iff, abelian, number, finite, subgroup, ordered group
There are 2 references to this entry.

This is version 9 of isolated subgroup, born on 2004-12-29, modified 2005-04-06.
Object id is 6605, canonical name is IsolatedSubgroup.
Accessed 1731 times total.

Classification:

AMS MSC:	06A05 (Order, lattices, ordered algebraic structures :: Ordered sets :: Total order)
	20F60 (Group theory and generalizations :: Special aspects of infinite or finite groups :: Ordered groups)

Pending Errata and Addenda

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