Weakly procharacteristic subgroup

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Definition

Definition with symbols

A subgroup $H$ of a group $G$ is termed weakly procharacteristic if for any automorphism $\sigma$ of $G$, the following holds: if $K$ denotes the closure of $H$ under the action of the cyclic group generated by $\sigma$, there exists $g\in K$ such that $\sigma (H)=gHg^{-1}$.

Relation with other properties

Stronger properties

• Characteristic subgroup
• Intermediately isomorph-conjugate subgroup
• Intermediately automorph-conjugate subgroup
• Procharacteristic subgroup

Weaker properties

• Automorph-conjugate subgroup
• Weakly pronormal subgroup

Effect of property operators

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The intermediately operator

Applying the intermediately operator to this property gives: intermediately automorph-conjugate subgroup

If $H$ is a subgroup of $G$ that is weakly procharacteristic in every intermediate subgroup of $G$ containing it, then $H$ is an intermediately automorph-conjugate subgroup of $G$. Conversely, if $H$ is intermediately automorph-conjugate in $G$, then $H$ is weakly procharacteristic in every intermediate subgroup.

Facts

• Any weakly procharacteristic subgroup of a normal subgroup is weakly pronormal. Further information: Weakly procharacteristic of normal implies weakly pronormal
• Weak procharacteristicity is the most general property for which this is true. Further information: Left residual of weakly pronormal by normal is weakly procharacteristic