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. A typical function restriction formal expression looks like:

$a\to b$

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

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.