Powering-injective group for a set of primes

Definition
Let $$\pi$$ be a set of primes. A group $$G$$ is termed $$\pi$$-powering-injective if it satisfies the following equivalent definitions:

Weaker properties

 * Torsion-free group for a set of primes

Other related properties

 * Powered group for a set of primes: Here, the powering maps need to be bijective.
 * Divisible group for a set of primes: Here, the powering maps need to be surjective.

Journal references

 * : The paper uses the notation U_{\pi}-group for this idea. The notation is introduced in Section 1 (Page 218, second page of the PDF).