# Power automorphism

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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 $\phi(H)\leq H$