Coprime automorphism-faithful subgroup

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Definition

A subgroup $H$ of a finite group $G$ is termed coprime automorphism-faithful if, given any non-identity automorphism $\sigma \in \operatorname {Aut} (G)$ such that $\sigma (H)=H$, and such that the order of $\sigma$ is relatively prime to the order of $G$, $\sigma$ acts nontrivially on $H$. Equivalently, any coprime automorphism group of $G$ that acts trivially on $G$ must be trivial.

Relation with other properties

• Coprime automorphism-faithful characteristic subgroup