Divisible group for a set of primes
Let be a set of primes. A group is termed -divisible if it satisfies the following equivalent definitions:
|1||-divisible, for each prime||For every and every , there exists such that . In other words, the map is surjective from to itself for all .|
|2||-divisible for every -number||if and is a natural number all of whose prime divisors are in the set , then there exists an element such that .|
|3||(not necessarily unique) rational powers with denominators -number||if and are integers with all prime divisors of in , there exists satisfying .|
- Powered group for a set of primes
- Powering-injective group for a set of primes
- Torsion-free group for a set of primes
Conjunction with group properties
- Nilpotent group that is divisible for a set of primes is the conjunction with the property of being a nilpotent group.
- Some aspects of groups with unique roots by Gilbert Baumslag, Acta mathematica, Volume 104, Page 217 - 303(Year 1960): PDF (ungated)More info: This paper uses the notation -group to describe what we call a -divisible group. The notation is introduced in Section 2, Page 218 (second page of the paper).