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 View subgroup properties expressible using this formalism

To see function restriction expressions for a number of subgroup properties, and some of the associated metaproperties of these, refer to the function restriction formalism chart

Definition

Main definition

A function restriction 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 $H$ satisfies the property in $G$ if and only if every function satisfying $a$ on $G$ restricts to a function satisfying $b$ in the set corresponding to $H$. (Note that in particular, every function satisfying property $a$ in $G$ should send $H$ to itself).

The property on the left of the arrow is termed the left side of the function restriction expression, and the property on the right side of the arrow is termed the right side of the function restriction expression.

Related formal expressions

• Function extension expression
• Subgroup intersection restriction 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 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 expression.

• Right tightening: This tries to find the strongest property $d$ such that $p=a\to d$. Here, $d$ is the property of being a function from a group $H$ to itself, such that there exists a group $G$ containing $H$ as a subgroup with property $p$, and a function satisfying $a$ in $G$, whose restriction to $H$ is the given function. The right tightening operation is idempotent, and a function restriction expression that arises as a result of right tightening is termed a right tight function restriction 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 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 expression, we need to have some function restriction expression to begin with.

Implication relations

Further information: Proving implications using function restriction expressions

If we have two subgroup properties:

$p=a\to b,q=c\to d$

and the condition $c\le a,b\le d$ are satisfied (in other words, any function satisfying property $c$ satisfies property $a$, and any function satisfying property $b$ also satisfies property $d$).

Then, $p\le q$. In other words, any subgroup satisfying property $p$ also satisfies $q$.

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 $p$.

This stronger result arises from the transiter master theorem.

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 $p$.

This stronger result arises from the transiter master theorem.

Particular kinds of function restriction expressions

Balanced expression

A function restriction expression is said to be balanced if the left side and the right side are equal. A subgrop property that possesses a balanced function restriction expression is termed a balanced subgroup property. Clearly, any balanced subgroup property must be a t.i. subgroup property.

Interestingly, the transiter master theorem gives us a partial converse: any function-restriction-expressible subgroup property, that is also a t.i. subgroup property, is actually balanced. This arises from the fact that it must equal its left transiter (or alternatively, its right transiter).

Invariance expression

An invariance expression is a function restriction expression where the right side is the tautology, or the property of being any function. A subgroup property that possesses an invariance expression is termed an invariance property. Basically an invariance property just means that the subgroups satisfying thep roperty are precisely those ones that are invariant under a certain set of functions.

Invariance properties are strongly intersection-closed as well as closed under unions of ascending chains of subgroups.