Difference between revisions of "Universal power map"

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m (Uniform power map moved to Universal power map over redirect)
 
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===Symbol-free definition===
 
===Symbol-free definition===
  
A '''uiniform power map''' or '''universal 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.
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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 '''uniform power map''' or '''universal 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>.
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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===
  
* [[Uniform power endomorphism]] is a universal power map that is also an [[endomorphism]]
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* [[Universal power endomorphism]] is a universal power map that is also an [[endomorphism]]
* [[Uniform power automorphism]] is a universal power map that is also an [[automorphism]]
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* [[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

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 f on a group G is termed a universal power map or uniform power map if there exists an integer n such that f(x) = x^n for all x in G.

Relation with other properties

Automorphisms and endomorphisms

For Abelian groups, all uniform power maps are endomorphisms.

Particular cases