Universal power map

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 $$G$$.

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