# Universal power map

From Groupprops

*This article defines a function property, viz a property of functions from a group to itself*

## Contents

## Definition

### Symbol-free definition

A **universal power map** or **uniform power map** is a function from a group to itself such that there exists an integer for which the function is simply raising to the power of that integer.

### Definition with symbols

A function on a group is termed a **universal power map** or **uniform power map** if there exists an integer such that for all in .

## Relation with other properties

### Automorphisms and endomorphisms

- Universal power endomorphism is a universal power map that is also an endomorphism
- Universal power automorphism is a universal power map that is also an automorphism

For Abelian groups, all uniform power maps are endomorphisms.