Template:Subgroup property

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under equivalence of subgroups.
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VIEW RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions | Subgroup metaproperty dissatisfactions | Subgroup property satisfactions |Subgroup property dissatisfactions
VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property
<random>RANDOM TIP:<random>The relation with other properties section lists stronger and weaker properties, along with links to proofs of the implications and non-implications. This helps give a feel of how the subgroup property relates to other properties.@@@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.@@@The testing section provides information on practical testing for the subgroup property, including implementation in GAP, when possible.</random>@@@RANDOM SUBGROUP PROPERTY: <random>Permutable subgroup: A subgroup that commutes with every other subgroup. May not be normal.@@@Hall subgroup: A subgroup of a finite group whose order and index are relatively prime. Sylow subgroups are a special case.@@@Automorph-conjugate subgroup: A subgroup of a group that is conjugate to any automorphic subgroup. Any characteristic subgroup is automorph-conjugate.@@@2-subnormal subgroup: A normal subgroup of a normal subgroup. Need not be normal, because normality is not transitive.@@@Fully characteristic subgroup: A subgroup such that any endomorphism of the whole group takes the subgroup to within itself.@@@Central factor: A subgroup with the property that every inner automorphism of the whole group restricts to an inner automorphism of the subgroup.@@@Abelian normal subgroup: A subgroup that is Abelian as a group and normal as a subgroup.@@@Contranormal subgroup: A subgroup whose normal closure in the whole group is the whole group.@@@Nearly normal subgroup: A subgroup having finite index inside its normal closure.@@@Conjugacy-closed subgroup: A subgroup such that any two elements of the subgroup that are conjugate in the whole group are conjugate in the subgroup.@@@Local subgroup: A subgroup that occurs as the normalizer of a nontrivial solvable subgroup.</random></random>

Below is the documentation for the template Template:Subgroup property

This template is an article-tagging template: it is intended to tag an entire article rather than a specific section.

Invoking the template:

• What articles are tagged by this template: Those articles that are about a subgroup property. A subgroup property is a property such that, given any group and any subgroup of that group, the group-sugbroup pair either satisfies the property or does not satisfy the property (a complete list of tagged articles is at Category:Subgroup properties).
• The template does not take in any parameters

Effects of the template:

• The box shown above appears on the article page
• The article automatically gets included in Category:Subgroup properties