# Group metaproperty

From Groupprops

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

## Contents |

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## Definition

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

## Examples

An example of a group metaproperty is being quotient-closed. A group property is termed quotient-closed or Q-closed if whenever a group satisfies the properties, so does every quotient of the group. Note that for any group property it either *is Q-closed* or *is not Q-closed*.