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 direct product-closed if it satisfies the following condition:

1. For any positive integer $n$ and groups $G_{1},G_{2},\dots$ all of which satisfy $\alpha$, the external direct product $G_{1}\times G_{2}\times \dots \times G_{n}$ also satisfies $\alpha$, as well as the infinite direct product $G_{1}\times G_{2}\times \dots$.

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

Relation with other metaproperties

Weaker metaproperties

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