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 definition
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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.