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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.

Relation with other properties

Weaker properties

Other related properties

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