# Power automorphism

From Groupprops

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 definitionVIEW: Definitions built on this | Facts about this: (factscloselyrelated 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

## Definition

### 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 , an automorphism is a power automorphism if the following equivalent conditions are satisfied:

- for each there exists a positive natural number such that
- for each we have

## Relation with other properties

### Stronger properties

- Uniform power automorphism:
*For proof of the implication, refer Uniform power automorphism implies power automorphism and for proof of its strictness (i.e. the reverse implication being false) refer Power automorphism not implies uniform power automorphism*. - Strong power automorphism: A strong power automorphism is a power automorphism whose inverse is also a power automorphism.