# 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 function from a group to itself is termed a **power map** if the following equivalent conditions hold:

- It takes each element to a power of that element
- It takes each subgroup to within itself

### Definition with symbols

A function from a group to itself is termed a **power map** if the following equivalent conditions hold:

- For any in , there exists an integer such that .
- For any subgroup of , and any element in , is also in .

## Relation with other properties

### Automorphisms and endomorphisms

- Power endomorphism is a power map that is also an endomorphism
- Power automorphism is a power map that is also an automorphism

### Stronger properties

A universal power map is a power map where we can fix the powering exponent independent of the element. That is, there is an integer such that for all in the group.