# Join-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]

This article is about a general term. A list of important particular cases (instances) is available at Category:Join-closed subgroup properties

## History

*This term is local to the wiki. To learn more about why this name was chosen for the term, and how it does not conflict with existing choice of terminology, refer the talk page*

## Definition

### Symbol-free definition

A subgroup property is termed **join-closed** if the join of a nonempty (but otherwise arbitrary, possibly infinite) collection of subgroups, each with property , also has property .

### Definition with symbols

A subgroup property is termed **join-closed** if given a group , a nonempty indexing set , and a collection of subgroups for , such that each satisfies , the join, i.e. the subgroup generated by the s, also satisfies .

## Relation with other metaproperties

### Stronger metaproperties

- Strongly join-closed subgroup property: This is a subgroup property that is both join-closed and trivially true
- Upward-closed subgroup property