Procharacteristic subgroup

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Definition

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 $H^{g}=H^{\sigma }$.

Relation with other properties

Stronger properties

• Characteristic subgroup
• Intermediately isomorph-conjugate subgroup

Weaker properties

• Automorph-conjugate subgroup
• Weakly procharacteristic subgroup
• Pronormal subgroup
• Weakly pronormal subgroup

Facts

• Any procharacteristic subgroup of a normal subgroup is pronormal. Further information: Procharacteristic of normal implies 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 $K$. Further information: Left residual of pronormal by normal is procharacteristic