# Subgroup metaproperty

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

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## 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 such that if and has the property in and has the property in , then has the property in . 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.