# Powering-injective group for a set of primes

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## 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 $n$.

## References

### Journal references

• 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 [itex]U_{\pi}-group for this idea. The notation is introduced in Section 1 (Page 218, second page of the PDF).