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 $n$.

Relation with other properties

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.