Universal power map: Difference between revisions

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

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.

Particular cases

@@ Line 5: / Line 5: @@
 ===Symbol-free definition===
-A '''uiniversal 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.
+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 <math>f</math> on a [[group]] <math>G</math> is termed a '''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>.
+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==
@@ Line 18: / Line 18: @@
 * [[Universal power automorphism]] is a universal power map that is also an [[automorphism]]
-For [[Abelian group]]s, all universal power maps are endomorphisms.
+For [[Abelian group]]s, all uniform power maps are endomorphisms.
 ==Particular cases==