Subgroup metaproperty

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This article is about a general term. A list of important particular cases (instances) is available at Category:Subgroup metaproperties


BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

Definition

A subgroup metaproperty is a map from the collection of all subgroup properties (viz., the subgroup property space) to the two-element set (true, false). Those subgroup properties which get mapped to true are said to have or possess the subgroup metaproperty, and those that map to false are said to not have or not possess the subgroup metaproperty.

Examples

An example of a subgroup metaproperty is transitivity. A transitive subgroup property is a subgroup property p such that if H \le K \le G and K has the property in G and H has the property in K, then H has the property in G. Transitivity is an important subgroup metaproperty, and many of the subgroup properties we encounter in practice are transitive. Many others are not, this itself is a subject of study.

Another example is identity-true subgroup property: an identity-true subgroup property is a subgroup property that is satisfied by every group, as a subgroup of itself. A trim subgroup property is a subgroup propery that is satisfied, in every group, by the whole group and the trivial subgroup.

Subgroup metaproperties often arise in practice as a result of binary subgroup property operators or subgroup property modifiers. For instance, the subgroup metaproperty of being transitive arises from the composition operator, and the subgroup metaproperty of being intersection-closed arises from the intersection operator.