# Difference between revisions of "Universal power map"

From Groupprops

m (Uniform power map moved to Universal power map over redirect) |
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===Symbol-free definition=== | ===Symbol-free definition=== | ||

− | A ''' | + | 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=== | ===Definition with symbols=== | ||

− | A function <math>f</math> on a [[group]] <math>G</math> is termed a ''' | + | A function <math>f</math> on a [[group]] <math>G</math> is termed a '''universal power map''' or '''uniform power map''' if there exists an integer <math>n</math> such that <math>f(x) = x^n</math> for all <math>x</math> in <math>G</math>. |

==Relation with other properties== | ==Relation with other properties== | ||

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===Automorphisms and endomorphisms=== | ===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 group]]s, all uniform power maps are endomorphisms. | For [[Abelian group]]s, all uniform power maps are endomorphisms. |

## Latest revision as of 10:57, 5 September 2008

*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.