# Rationally powered group

## Contents

## 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 propertiesVIEW 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 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 |

## 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.