# Left-hereditary 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: Left-hereditary subgroup properties

## Definition

### Symbol-free definition

A subgroup property is said to be **left-hereditary** if any subgroup of a subgroup having the property, also has the property.

### Definition with symbols

A subgroup property is said to be **left-hereditary** if given groups ≤ ≤ , such that satisfies property in , also satisfies property in .

### In terms of the left transiter

A subgroup property is termed **left-hereditary** if its left transiter is the tautology.