Groupprops, The Group Properties Wiki (pre-alpha)

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Contents 1 Definition 1.1 Definition with symbols 2 Relation with other properties 2.1 Stronger properties 2.2 Weaker properties 3 Facts

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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 SUBGROUP PROPERTY: Abelian normal subgroup: A subgroup that is Abelian as a group and normal as a subgroup.

Definition

Definition with symbols

• (Left-action convention): A subgroup H of a group G is termed procharacteristic in G if, for any automorphism σ of G, there exists such that gHg ^{− 1} = σ(H).
• (Right-action convention): A subgroup H of a group G is termed procharacteristic in G if, for any automorphism σ of G, there exists such that H^g = H^σ.

Relation with other properties

Stronger properties

• Characteristic subgroup
• Intermediately isomorph-conjugate subgroup

Weaker properties

• Automorph-conjugate subgroup
• Weakly procharacteristic subgroup
• Pronormal subgroup
• Weakly pronormal subgroup

Facts

• Any procharacteristic subgroup of a normal subgroup is pronormal. Further information: Procharacteristic of normal implies pronormal
• A subgroup H of a group G is procharacteristic in G if and only if whenever G is normal in some group K, H is pronormal in K. Further information: Left residual of pronormal by normal is procharacteristic