# Group in which every power map is injective

From Groupprops

## Contents

## Definition

A group is termed a **group in which every power map is injective** or a **R-group** or a **powering-injective group** if it satisfies the following equivalent conditions:

- For every prime number , the power map is injective from to itself.
- For every natural number , the power map is injective from to itself.
- For every nonzero integer , the power map is injective from to itself.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

rationally powered group | every power map is bijective |
(obvious) | the group of integers is an example. | |FULL LIST, MORE INFO |

group embeddable in a rationally powered group | Powering-injective group need not be embeddable in a rationally powered group |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

torsion-free group | no non-identity elements of finite order | (obvious) | torsion-free not implies every power map is injective |