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
A group property is termed finite direct product-closed if it satisfies the following equivalent conditions:
- Whenever and are groups satisfying , the external direct product also satisfies .
- For any positive integer and groups all of which satisfy , the external direct product also satisfies .
Note that if the trivial group also satisfies , we say that is strongly finite direct product-closed.
Relation with other 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|