isolated subgroup
Let be a ordered group and its subgroup. We call this subgroup if every element of and every element of satisfy
If an ordered group has only a finite number of isolated subgroups, then the number of proper () isolated subgroups of is the of .
Theorem.
Let be an abelian ordered group with order (http://planetmath.org/OrderGroup) at least 2. The of equals one iff there is an order-preserving isomorphism
from 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).