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From Groupprops

Contents 1 Definition 2 Relation with other properties 2.1 Stronger properties 2.2 Weaker properties 3 Facts 4 Metaproperties 4.1 Trimness

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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.
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RANDOM TIP:The metaproperties section lists important facts about the subgroup property, and addresses many of the natural questions that arise about it. It has links to proofs.

Definition

A subgroup H of a group G is termed polycharacteristic in G if the following holds: for any automorphism σ of G, H is a contranormal subgroup in the closure of H in G under the action of the cyclic subgroup generated by σ.

Relation with other properties

Stronger properties

• Characteristic subgroup
• Procharacteristic subgroup
• Weakly procharacteristic subgroup
• Paracharacteristic subgroup
• Intermediately isomorph-conjugate subgroup
• Intermediately automorph-conjugate subgroup

Weaker properties

• Normal-to-characteristic subgroup

Facts

• Polycharacteristic of normal implies polynormal
• Left residual of polynormal by normal equals polycharacteristic

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 all trim subgroup properties OR view trivially true subgroup properties OR view identity-true subgroup properties