# Left-extensibility-stable subgroup property

This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property
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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 $p$ is termed left-extensibility-stable if we can write:

$p = a \to b$

where $a$ is an extensibility-stable function property.

The above symbols mean that:

• A subgroup $H$ has property $p$ in a group $G$ if and only if every function from $G$ to itself satisfying property $a$ restricts to a function from $H$ to itself satisfying property $b$.
• Property $a$ being extensibility-stable means the following: whenever $G \le K$ are groups,

and $f$ is a function on $G$ satisfying $a$, then there is a function $f'$ on $K$ satisfying $a$ such that the restriction of $f'$ to $G$ is $f$.

### 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.