Groupprops:Relation with other properties

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This article provides information on a particular section found in some Groupprops articles

The Relation with other properties section is found in Groupprops property definition articles. The purpose of this section is to relate different properties in the same property space (That is, evaluated over the same collection of objects). For instance, this section for a subgroup property gives other related subgroup properties.

We use two parameters of relatedness:

• What implies what: If every object satisfying property $p$ also satisfies property $q$, then we say that $p$ is stronger than $q$, and that $q$ is weaker than $p$

• How closely the definitions are related structurally

Stronger and weaker properties

The Relation with other properties section for any property definitino article contains these two subsections (among others):

• The Stronger properties subsection. This gives, as an unordered list, some properties that are stronger than the property being studied by the article. Proofs and explanations are omitted, though a link to a proof may be given. Further information: Groupprops:Property implication

• The Weaker properties subsection. This gives, as an unordered list, some properties that are weaker than the property being studied by the article. Proofs and explanations are omitted, though a link to a proof may be given. Further information: Groupprops:Property implication

Note that we do not try to be exhaustive in the Stronger properties and Weaker properties sections. In other words, we do not list all the properties which are stronger, or weaker, than the given property. But we try to list all the properties that are immediately or closely stronger. In other words, we list enough stronger and weaker properties such that if $p$ is stronger than $q$, we should be able to find a weaker property in $p$'s list, then a weaker property in the list of that property, and so on, to hit $q$.

Variations and opposites

For variations and opposites, explicit lists are not provided in the article itself, rather, links are provided to the appropriate categories listing variations, and opposites, of the given property.

Related properties

There may be other closely related, but incomparable, properties. These are listed in a separate subsection titled Related properties.