Universal power map

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

• Identity map
• Inverse map
• Square map
• Cube map