Paracharacteristic subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed paracharacteristic in $$G$$ if for any automorphism $$\sigma$$ of $$G$$, $$H$$ is a contranormal subgroup of the subgroup $$\langle H, \sigma(H) \rangle$$.

Stronger properties

 * Weaker than::Intermediately isomorph-conjugate subgroup
 * Weaker than::Procharacteristic subgroup
 * Weaker than::Characteristic subgroup
 * Weaker than::Abnormal subgroup
 * Weaker than::Weakly abnormal subgroup

Weaker properties

 * Stronger than::Paranormal subgroup
 * Stronger than::Polycharacteristic subgroup
 * Stronger than::Polynormal subgroup
 * Stronger than::Normal-to-characteristic subgroup

Facts

 * Paracharacteristic of normal implies paranormal
 * Left residual of paranormal by normal equals paracharacteristic