# Pseudovarietal group property

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

## Definition

A group property $\alpha$ is termed pseudovarietal if it satisfies the following three conditions:

1. It is a subgroup-closed group property, i.e., whenever $G$ is a group satisfying $\alpha$ and $H$ is a subgroup of $G$, $H$ also satisfies $\alpha$.
2. It is a quotient-closed group property, i.e., whenever $G$ is a group satisfying $\alpha$ and $H$ is a normal subgroup of $G$, the quotient group $G/H$ also satisfies $\alpha$.
3. It is a finite direct product-closed group property, i.e., whenever $G_1,G_2,\dots,G_n$ are groups all of which satisfy $\alpha$, the external direct product $G_1 \times G_2 \times \dots \times G_n$ also satisfies $\alpha$.

## 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