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.