Function restriction expression

This page describes a formal expression, or formalism, that can be used to describe certain subgroup properties.

View a complete list of formal expressions for subgroup properties OR [[:Category:{{{1}}}|View subgroup properties expressible using this formalism]]

Definition

Main definition

A function restriction formal expression is the expression of a subgroup property using the function restriction formalism. The function restriction formalism corresponding to function properties $a$ and $b$ is denoted as:

$a\to b$

meaning that every function satisfying $a$ on $G$ restricts to a function satisfying $b$ in the set corresponding to $H$.

Related formal expressions

• Function extension formal expression
• Subgroup intersection restriction formal expression

Expressing subgroup properties this way

Subgroup properties that can be expressed

A subgroup property that can be expressed via a function restriction formal expression is termed a function-restriction-expressible subgroup property. A list of all the subgroup properties that are function-restriction-expressible can be found at: Category:Function-restriction-expressible subgroup properties.

Composition operator

Composition rule

Let $p=a\to b$ and $q=c\to d$ be subgroup properties. Then if $d\le a$, we have:

$p*q\le c\to b$

For full proof, refer: composition rule for function restriction

Corollary for left transiter

Let $p=a\to b$ be a subgroup property. Then, if $q=b\to b$, $q*p\le p$.

This in particular means that the left transiter for $p$ is weaker than $q$. In fact, a stronger result holds: if $a\to b$ is a right tight restriction formal expression for $p$ (that is, $b$ cannot be strengthened further) then $q=b\to b$ is precisely the left transiter of $p$.

An example is where $p$ is the property of being normal. Setting $a$ as the property of being an inner automorphism and $b$ as the property of being an automorphism gives a right tight restriction formal expression for $p$. Hence, the left transiter is the property with both left side and right side being the property of being an automorphism. This is the subgroup property of being characteristic.

Corollary for right transiter

Let $p=a\to b$ be a subgroup property. Then, if $q=a\to a$, $p*q\le p$.

This in particular means that the right transiter for $p$ is weaker than $q$. In fact, a stronger result holds: if $a\to b$ is a left tight restriction formal expression]] for $p$, and $a\le b$, then $q=a\to a$ is precisely the right transiter of $p$.