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 in terms of two properties of functions (by function here is meant a function from a group to itself). The function restriction formal expression corresponding to function properties $a$ and $b$ is denoted as:

$a\to b$

meaning that $H$ satisfi es the property in $G$ if 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 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.

Canonical forms for expressing a given subgroup property

If we are given a function restriction formal expression $p=a\to b$, we can do two operations:

• Left tightening: This tries to find the weakest property $c$ such that $p=c\to b$. Here, $c$ is the property of being a function from a group $H$ to itself that restricts to a function satisfying property $b$ in every subgroup $H$ satisfying property $p$ in $G$. The left tightening operation is idempotent, and a function restriction formal expression that arises as a result of left tightening is termed a left tight function restriction formal expression.

• Right tightening: This tries to find the strongest property $d$ such that $p=a\to d$. Here, $d$ is the proeprty of being a function from a group $H$ to itself, such that there exists a group $G$ containing $H$ and a function satisfying $a$ in <mtah>G</math>, whose restriction to $H$ is the given function. The right tightening operation is idempotent, and a function restriction formal expression that arises as a result of right tightening is termed a right tight function restriction formal expression.

There are some implicit assertions made in the above definitions which are not hard to justify.

If a subgroup property is function-restriction-expressible, then it possesses both a left tight and a right tight function restriction formal expression, by the above logic. Further, right tightening preserves left tightness, so if we apply both the left and the right tightening operations, we get a property that is both left and right tight. However, it seems that the order in which we apply the left and right tightening operations, could affect the final answer we get.

Notice, however, that to be able to obtain a left tight and/or a right tight function restriction formal expression

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 $p$.