Procharacteristic subgroup

Definition with symbols

 * (Left-action convention): A subgroup $$H$$ of a group $$G$$ is termed procharacteristic in $$G$$ if, for any automorphism $$\sigma$$ of $$G$$, there exists $$g \in \langle H, \sigma(H) \rangle$$ such that $$gHg^{-1} = \sigma(H)$$.
 * (Right-action convention): A subgroup $$H$$ of a group $$G$$ is termed procharacteristic in $$G$$ if, for any automorphism $$\sigma$$ of $$G$$, there exists $$g \in \langle H, H^\sigma\rangle$$ such that $$H^g = H^\sigma$$.

Stronger properties

 * Weaker than::Characteristic subgroup
 * Weaker than::Intermediately isomorph-conjugate subgroup

Weaker properties

 * Stronger than::Automorph-conjugate subgroup
 * Stronger than::Weakly procharacteristic subgroup
 * Stronger than::Pronormal subgroup
 * Stronger than::Weakly pronormal subgroup

Facts

 * Any procharacteristic subgroup of a normal subgroup is pronormal.
 * A subgroup $$H$$ of a group $$G$$ is procharacteristic in $$G$$ if and only if whenever $$G$$ is normal in some group $$K$$, $$H$$ is pronormal in $$K$$.