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

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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 $G\leq K$ 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.

Relation with other metaproperties

Stronger metaproperties

• Extensibility-stable invariance property
• Left-inner subgroup property

Weaker metaproperties

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