# 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 metaproperty
VIEW RELATED: subgroup metaproperty satisfactions| subgroup metaproperty dissatisfactions

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

## 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.