Procharacteristic subgroup

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Definition

Definition with symbols

A subgroup $H$ of a group $G$ is termed procharacteristic in $G$ if, for any automorphism $σ$ of $G$ , there exists $g \in ⟨ H, σ (H) ⟩$ such that $g H g^{- 1} = σ (H)$ .

Relation with other properties

Stronger properties

Intermediately isomorph-conjugate subgroup

Weaker properties

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