# Finite-intersection-closed subgroup property

This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property

View a complete list of subgroup metaproperties

View subgroup properties satisfying this metaproperty| View subgroup properties dissatisfying this metapropertyVIEW RELATED: subgroup metaproperty satisfactions| subgroup metaproperty dissatisfactions

BEWARE!This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

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 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.