Power map

This article defines a function property, viz a property of functions from a group to itself

Definition

Symbol-free definition

A function from a group to itself is termed a power map if the following equivalent conditions hold:

• It takes each element to a power of that element
• It takes each subgroup to within itself

Definition with symbols

A function $f$ from a group $G$ to itself is termed a power map if the following equivalent conditions hold:

• For any $x$ in $G$, there exists an integer $n$ such that $f(x)=x^{n}$.
• For any subgroup $H$ of $G$, and any element $x$ in $H$, $f(x)$ is also in $H$.

Relation with other properties

Automorphisms and endomorphisms

• Power endomorphism is a power map that is also an endomorphism
• Power automorphism is a power map that is also an automorphism

Stronger properties

A universal power map is a power map where we can fix the powering exponent independent of the element. That is, there is an integer $n$ such that $f(x)=x^{n}$ for all $x$ in the group.