Finite direct product-closed 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 finite direct product-closed if it satisfies the following equivalent conditions:

1. Whenever $G_1$ and $G_2$ are groups satisfying $\alpha$, the external direct product $G_1 \times G_2$ also satisfies $\alpha$.
2. For any positive integer $\alpha$ and groups $G_1,G_2,\dots,G_n$ all of which satisfy $\alpha$, the external direct product $G_1 \times G_2 \times \dots \times G_n$ also satisfies $\alpha$.

Note that if the trivial group also satisfies $\alpha$, we say that $\alpha$ is strongly finite direct product-closed.

Relation with other metaproperties

Stronger metaproperties

Metaproperty	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
direct product-closed group property				|FULL LIST, MORE INFO
varietal group property				Pseudovarietal group property|FULL LIST, MORE INFO
quasivarietal group property				|FULL LIST, MORE INFO
pseudovarietal group property				|FULL LIST, MORE INFO