Power map

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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.