# Pseudovarietal group property

From Groupprops

This article defines a group metaproperty: a property that can be evaluated to true/false for any group property

View a complete list of group metaproperties

## Contents

## Definition

A group property is termed **pseudovarietal** if it satisfies the following three conditions:

- It is a subgroup-closed group property, i.e., whenever is a group satisfying and is a subgroup of , also satisfies .
- It is a quotient-closed group property, i.e., whenever is a group satisfying and is a normal subgroup of , the quotient group also satisfies .
- It is a finite direct product-closed group property, i.e., whenever are groups all of which satisfy , the external direct product also satisfies .

## Relation with other metaproperties

### Stronger metaproperties

Metaproperty | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

varietal group property | subgroup-closed, quotient-closed, and closed under arbitrary direct products |
|FULL LIST, MORE INFO |

### Weaker metaproperties

Metaproperty | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

subgroup-closed group property | closed under taking subgroups | |FULL LIST, MORE INFO | ||

quotient-closed group property | closed under taking quotient groups | |FULL LIST, MORE INFO | ||

finite direct product-closed group property | closed under taking finite direct products | |FULL LIST, MORE INFO |