|
|
|
|
![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
isolated subgroup
|
(Definition)
|
|
|
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 $G$ .
- 1
- M. LARSEN & P. MCCARTHY: Multiplicative theory of ideals. Academic Press. New York (1971).
|
"isolated subgroup" is owned by pahio.
|
|
(view preamble | get metadata)
Cross-references: real numbers, multiplicative group, onto, isomorphism, order-preserving, iff, abelian, number, finite, element, subgroup, ordered group
There are 2 references to this entry.
This is version 10 of isolated subgroup, born on 2004-12-29, modified 2009-02-22.
Object id is 6605, canonical name is IsolatedSubgroup.
Accessed 2487 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
|
|
|
|
|
|
|
|
|
|
|
