Groupprops, The Group Properties Wiki (pre-alpha)

YOUR FEEDBACK IS IMPORTANT!

Please take a short user satisfaction survey about Groupprops.

Your survey responses will be helpful in improving the site experience!

Thanks in advance!

From Groupprops

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

Contents 1 Definition 1.1 Symbol-free definition 1.2 Definition with symbols 2 Relation with other properties 2.1 Automorphisms and endomorphisms 2.2 Stronger properties

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) = xⁿ.
• 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) = xⁿ for all x in the group.