# Group powered over a unital ring

From Groupprops

## Contents

## Definition

Suppose is a group and is a unital ring. A structure of as a **group powered over** includes an operation denoted by exponentiation, i.e., the output of is denoted , satisfying the following conditions:

- 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.

## Functors

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 .

## Particular cases

### 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 |

### Unipotent algebraic groups

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