Group powered over a unital ring
- For an integer (modulo whatever is the characteristic of the ring), is the usual .
- For , .
- For ,
As a variety of algebras
For any fixed unital ring , the groups powered over form a variety of algebras. This variety admits the variety of groups as a reduct, i.e., every group powered over a unital ring gives rise to a group.
Suppose and are unital rings and is a homomorphism of unital rings. Then, if is a group equipped with a powering structure over , we naturally get a powering structure of over .
Cases based on rings
|Value of ring||Notion of group powered over that ring||Is the powering uniquely determined by the abstract group structure?||Does every group admit a powering over such a ring? Note that this condition would hold if the ring has a surjective ring homomorphism to .|
|usual concept of group||Yes||Yes|
|where is a set of primes||group powered over a set of primes where the set of primes is . In other words, every element of the group has a unique root in the group.||Yes||No (unless is empty)|
|rationally powered group: unique roots for all primes||Yes||No|
|for a positive integer||group whose exponent is finite and divides||Yes||No|
|group along with a set map from the group to itself (need not be a homomorphism) that commutes with powering and sends every element to an element that it commutes with.||No||Yes|
|rationally powered group along with a set map from the group to itself that commutes with powering and sends every element to an element that commutes with it.||No||No|