Groupprops:Property definition article

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This article is about the groupprops wiki itself

A property definition article is a definition article for a property.

All articles may not match the standards given here because these standards have themselves evolved along with groupprops. The article on normal subgroup shall be a prototype that shall always conform to the latest format, layout and standards.


Adjectival naming

Typically, a property is given by an adjective, followed by the noun that it qualifies (which denotes the context space over which it is a property). For instance the term normal subgroup comprises two words, the adjective normal and the noun subgroup which describes the space over which we are considering the notion of normal.

For adjectival naming, we use the adjective followed by the noun it qualifies as the article name.

Member-noun naming

Here, the property is given by a noun, where the noun would be used for a typical element having the property. For instance, the term direct factor denotes a subgroup property, viz and subgroup satisfying the proeprty is called a direct factor.

We simply use the member noun as the article name.

Property-noun naming

Here, the property is given by a noun, where any element satisfying the property is said to satisfy the property noun. For instance, the term normality is a property noun for the property of being normal, and any subgroup which is normal is said to satisfy the property of normality.

We simply use the property noun as the article name.

By and large, adjectival naming is preferred, so if both adjectival naming and member/property noun naming is used, redirect from the member/property noun article to the adjectival naming article.

Article-tagging templates

  • Use the appropriate property space specification template. A listing of property space specification templates is provided at:

Category: Property space specification templates

  • Check whether the term being defined is a variation of (in which case the template Template:Variationof is to be used) or an opposite of (in which case the template Template:Oppositeof is to be used) some more pivotal property.

Further information: Groupprops:Definition article-tagging templates

Parts of the property definition article

Property definition articles typically have the following sections:

  • History section which gives a brief account of the origin and subsequent development of the concept and term
  • Definition section where one or more definitions may be given
  • Formalisms section is a section where formalisms (or calculi, or formal descriptions) for the given property are described
  • Relation with other properties section is a section where the relation between the given property and other properties is discussed. This includes a list of stronger properties, a list of weaker properties, and a list of other closely related properties.

It may also include links to categories listing related properties.


Follow the same guidelines as for a general definition, which are specified in the Groupprops:Definition article.

Relation with other properties

The section relation with other properties is intended to give some idea of where the given property lies, both in relation with other properties in the same property space, and in relation with properties/operators/elements outside.

If the property is pivotal, the section begins with the use of Template:Pivotalproperty followed by a listing of the variation, opposites and other related categories. Subsections include:

  • Stronger properties: Here, the stronger properties are stated in a list and for each one, a link to the proof may be provided if it exists
  • Conjunction with other properties: Here, both the property with which conjunction is taken and the result of applying conjunction are specified
  • Weaker properties: Here, the weaker properties are stated in a list and for each one, a link to the proof may be provided if it exists
  • Related properties in other property spaces
  • Related operators


For the given property, we can, for every metaproperty, ask whether the property at hand satisfies the given metaproperty. Under the metaproperties section, we devote one subsection to each metaproperty. Each metaproperty will typically have a template that generates a section header as well as a clear yes/no for whether the metaproperty is satisfied.

The subsection should try to give a brief reason why the metaproperty is satisfied or not satisfied and a link to the article giving a proof/counterexample.

Note that the template for yes will also include the given property in the category of properties satisfying the given subgroup metaproperty.

The list of templates is available separately for each property space, and the list of these lists is at:

Category: Metaproperty satisfaction template types

Property operators

This section describes the effect of various property operators and proeprty modifiers on the given property. For each property operator, there is a corresponding template which writes an emphasized sentence on top saying that the effect of applying this modifier or operator is so-and-so.

The further content of the subsection provides a closer explanation of what is meant, and a link to the proof.


The testing section describes any related computational problem and provides a Template:Further link for more information on the computational problem.

For properties that can be tested using GAP implementations, the GAP command is provided. Check out for the appropriate template to use.