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)
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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 $G$, 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

• 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.

Weaker properties

• Power endomorphism
• Monomial automorphism
• Weakly normal automorphism