# Left-extensibility-stable subgroup property

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 subgroup metaproperty is related to, or can be defined, using the following formalism:* function restriction formalism

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-extensibility-stable subgroup properties

## Definition

### Symbol-free definition

A subgroup property is termed **left-extensibility-stable** if it can be expressed in the function restriction formalism with the left side being extensibility-stable.

### Definition with symbols

A subgroup property is termed **left-extensibility-stable** if we can write:

where is an extensibility-stable function property.

The above symbols mean that:

- A subgroup has property in a group if and only if every function from to itself satisfying property restricts to a function from to itself satisfying property .
- Property being extensibility-stable means the following: whenever are groups,

and is a function on satisfying , then there is a function on satisfying such that the restriction of to is .

### In terms of the left expressibility operator

The metaproperty of being a left-extensibility-stable subgroup property is obtained by applying the left expressibility operator to the function metaproperty of being extensibilility-stable.

## Relation with other metaproperties

### Stronger metaproperties

### Weaker metaproperties

- Intermediate subgroup condition:
`For full proof, refer: Left-extensibility-stable implies intermediate subgroup condition`