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

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.