Finite direct product-closed group property
From Groupprops
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 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
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 |