Power map

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 $$H$$.

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.