# Strongly intersection-closed subgroup property

From Groupprops

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]

## Definition

### Symbol-free definition

A subgroup property is said to be **strongly intersection-closed** if given any arbitrary (possibly empty) family of subgroups each satisfying the subgroup property in the group, the intersection of all the subgroups again satisfies the property.

### Definition with symbols

A subgroup property is termed **strongly intersection-closed** if given any group and any (possibly empty) family of subgroups of indexed by such that each satisfies in , the group also satisfies in .

In other words, a subgroup property is strongly intersection-closed if it is both intersection-closed and identity-true.

## Relation with other metaproperties

### Stronger metaproperties

- Strongly UL-intersection-closed subgroup property
- Invariance property:
`For full proof, refer: Invariance implies strongly intersection-closed`