Power automorphism

From Groupprops
Jump to: navigation, search
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

This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
VIEW: Definitions built on this | Facts about this: (facts closely related to Power automorphism, all facts related to Power automorphism) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
View a complete list of semi-basic definitions on this wiki


Symbol-free definition

An automorphism of a group is termed a power automorphism if it satisfies the following equivalent conditions:

  • It is a power map in the sense that it takes each element to a power of itself
  • It takes each subgroup to within itself.

Definition with symbols

Given a group G, an automorphism \phi\in \text{Aut}(G) is a power automorphism if the following equivalent conditions are satisfied:

  • for each g\in G there exists a positive natural number k such that \phi(g)=g^k
  • for each H\leq G we have \phi(H)\leq H

Relation with other properties

Stronger properties

Weaker properties