Finite-intersection-closed subgroup property
This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property
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This article is about a general term. A list of important particular cases (instances) is available at Category: Finite-intersection-closed subgroup properties
A subgroup property is termed finite-intersection-closed if an intersection of a finite nonempty collection of subgroups, each with property , also has property .
Definition with symbols
A subgroup property is termed finite-intersection-closed if whenever are subgroups of , each satisfying in , then the intersection of all the s also satisfies in .
In terms of the intersection operator
The binary intersection operator takes two subgroup properties and gives the property of being a subgroup obtained as the intersection of subgroups with these properties. The intersection operator is a commutative associative quantalic binary operator, and the property of being finite-intersection-closed is the property of being transitive with respect to this operator.
Relation with other metaproperties
A slight change in definition
We define being strongly finite-intersection-closed as being both finite-intersection-closed and identity-true. This is the same as being t.i. with respect to the intersection operator.