Paracharacteristic subgroup

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

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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 $\langle H,\sigma (H)\rangle$.

Relation with other properties

Stronger properties

• Intermediately isomorph-conjugate subgroup
• Procharacteristic subgroup
• Characteristic subgroup
• Abnormal subgroup
• Weakly abnormal subgroup

Weaker properties

• Paranormal subgroup
• Polycharacteristic subgroup
• Polynormal subgroup
• Normal-to-characteristic subgroup

Facts

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

Metaproperties

Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties