# Upward-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

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

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 **upward-closed** or sometimes, **right-hereditary**, if whenever a subgroup has the property in a group, every intermediate subgroup also has the property in the group.

### Definition with symbols

A subgroup property is termed **upward-closed** or sometimes **right-hereditary** if whenever ≤ ≤ are groups such that satisfies property in , then also satisfies property in .

### In terms of the upward closure operator

A subgroup property is upward-closed if and only if it is unchanged under application of the upward closure operator. The upward closure operator is an idempotent property operator for subgroup properties.