Varietal 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 varietal 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 direct product-closed group property, i.e., whenever $G_{1},G_{2},\dots$ are groups all of which satisfy $\alpha$, the external direct product $G_{1}\times G_{2}\times \dots \times G_{n}$ for any $n$, and the infinite direct product $G_{1}\times G_{2}\times \dots$ also satisfy $\alpha$.

Relation with other metaproperties

Weaker metaproperties

Metaproperty	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
pseudovarietal group property	subgroup-closed, quotient-closed, and closed under finite direct products			|FULL LIST, MORE INFO
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
direct product-closed group property	closed under taking direct products			|FULL LIST, MORE INFO