Powering-injective group for a set of primes
From Groupprops
Contents
Definition
Let be a set of primes. A group is termed -powering-injective if it satisfies the following equivalent definitions:
No. | Shorthand | Explanation |
---|---|---|
1 | -powering-injective, for each prime | For every and every , there exists at most one value such that . In other words, the map is injective from to itself for all . |
2 | -powering-injective for every -number | if and is a natural number all of whose prime divisors are in the set , then there exists at most one element satisfying . In other words, the power map is injective for all -numbers . |
Relation with other properties
Weaker 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.
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).