Rationally powered group
A group is termed rationally powered or uniquely divisible if it satisfies the following equivalent conditions:
- For every and every natural number , there is a unique such that .
- is a powered group for all prime numbers.
- For any integers with , and for any , there exists a unique such that .
More generally, we can talk of a powered group for a set of primes.
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
Relation with other properties
|Property||Meaning||Proof of implication||Proof of strictness (reverse implication failure)||Intermediate notions|
|torsion-free group||no non-identity element of finite order||obvious||the group of integers is torsion-free but not rationally powered.||Group in which every power map is injective|FULL LIST, MORE INFO|
|divisible group||every element has at least one root for every||obvious (rationally powered additionally guarantees uniqueness)||the group of rational numbers modulo integers|||FULL LIST, MORE INFO|
|powering-injective group||no two different elements can have the same power.||obvious||the group of integers is powering-injective but not rationally powered.|||FULL LIST, MORE INFO|
- A nilpotent group is rationally powered iff it is divisible and torsion-free. This follows from the more general fact that in a nilpotent torsion-free group, if roots exist, they are unique.