Finite-intersection-closed 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

Definition

Symbol-free definition

A subgroup property $p$ is termed finite-intersection-closed if an intersection of a finite nonempty collection of subgroups, each with property $p$, also has property $p$.

Definition with symbols

A subgroup property $p$ is termed finite-intersection-closed if whenever $H_{1},H_{2},...,H_{n}$ are subgroups of $G$, each satisfying $p$ in $G$, then the intersection of all the $H_{i}$s also satisfies $p$ in $G$.

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.

Stronger metaproperties

• Strongly intersection-closed subgroup property
• Intersection-closed subgroup property
• Strongly finite-intersection-closed subgroup property
• Transitive subgroup property satisfying transfer condition: For full proof, refer: Transitive and transfer condition implies finite-intersection-closed