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:

No.	Shorthand	Explanation
1	$p$ -powering-injective, for each prime $p\in \pi$	For every $g\in G$ and every $p\in \pi$ , there exists at most one value $h\in G$ such that $h^{p}=g$ . In other words, the map $x\mapsto x^{p}$ is injective from $G$ to itself for all $p\in \pi$ .
2	$n$ -powering-injective for every $\pi$ -number $n$	if $g\in G$ and $n$ is a natural number all of whose prime divisors are in the set $\pi$ , then there exists at most one element $h\in G$ satisfying $h^{n}=g$ . In other words, the $n^{th}$ power map is injective for all $\pi$ -numbers $n$ .

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.

Some aspects of groups with unique roots by Gilbert Baumslag, Acta mathematica, Volume 104, Page 217 - 303(Year 1960): ^{PDF (ungated)}^{More info}: The paper uses the notation <math>U_{\pi}-group for this idea. The notation is introduced in Section 1 (Page 218, second page of the PDF).