Powering-injective group for a set of primes
Let be a set of primes. A group is termed -powering-injective if it satisfies the following equivalent definitions:
|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
- 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.