Universal power automorphism

This article defines an automorphism property, viz a property of group automorphisms. Hence, it also defines a function property (property of functions from a group to itself)
View other automorphism properties OR View other function properties

Definition

Symbol-free definition

An automorphism of a group is termed a universal power automorphism or uniform power automorphism if it is also a universal power map: it can be viewed as taking the $n^{th}$ power for some integer $n$.

Definition with symbols

An automorphism $\sigma$ of a group $G$ is termed a universal power automorphism if there exists an integer $n$ such that $\sigma (g)=g^{n}$ for all $g\in G$.

Relation with other properties

Weaker properties

• Power automorphism: For proof of the implication, refer Universal power automorphism implies power automorphism and for proof of its strictness (i.e. the reverse implication being false) refer Power automorphism not implies universal power automorphism.