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 $n$ 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				|FULL LIST, MORE INFO
quasivarietal group property				|FULL LIST, MORE INFO
pseudovarietal group property				|FULL LIST, MORE INFO