Rationally powered group
From Groupprops
Definition
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
Weaker 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 ![]() ![]() |
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 ![]() |
obvious | the group of integers is powering-injective but not rationally powered. | |FULL LIST, MORE INFO |
Facts
- 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.