Group in which every power map is injective

Definition

A group $G$ 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:

1. For every prime number $p$, the power map $x\mapsto x^p$ is injective from $G$ to itself.
2. For every natural number $n$, the power map $x\mapsto x^n$ is injective from $G$ to itself.
3. For every nonzero integer $n$, the power map $x\mapsto x^n$ is injective from $G$ 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