Groupprops, The Group Properties Wiki (pre-alpha)

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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VIEW FACTS THAT:| prove, or show that, a subgroup satisfies the property
RANDOM SUBGROUP PROPERTY: Permutable subgroup: A subgroup that commutes with every other subgroup. May not be normal.

Definition

A subgroup H of a group G is termed paracharacteristic in G if for any automorphism σ of G, H is a contranormal subgroup of the subgroup .

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