Finite-intersection-closed subgroup property

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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]
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_is 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