Rational class-preserving 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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Definition

An automorphism $\sigma$ of a group $G$ is termed a rational class-preserving automorphism if it satisfies the following equivalent conditions:

• For any $g\in G$, there exists $h\in G$ such that $\langle \sigma (g)\rangle$ and $\langle hgh^{-1}\rangle$ are equal.
• $\sigma$ sends every cyclic subgroup to a conjugate subgroup.

Relation with other properties

Stronger properties

• Power automorphism
• Monomial automorphism
• Inner automorphism
• Class-preserving automorphism
• Extended class-preserving automorphism
• Subgroup-conjugating automorphism

Weaker properties

• Normal automorphism
• Weakly normal automorphism