Subgroup property modifier

This article is about the notion of property modifier for the [[{{{1}}} property space]]: it inputs a [[{{{1}}} property]] and outputs a [[{{{1}}} property]]

This article is about a general term. A list of important particular cases (instances) is available at Category:Subgroup property modifiers

Definition

Symbol-free definition

A subgroup property modifier is a function from the subgroup property space to itself that takes as input a subgroup property and outputs a subgroup property.

Examples

An example of a subgroup property modifier is the subordination operator, which takes as input a subgroup property $p$, and outputs the property of being a subgroup, which can be connected to the whole group via a series of intermediate subgroups, each having property $p$ in the next. This is the Kleene-star closure with respect to the composition operator for subgroup properties.

Another example is the left transiter of a subgroup property. The left transiter of a subgroup property $p$ is the following subgroup property $q$: $H$ has property $q$ in $G$ if whenever $G \le K$ such that $G$ has property $p$ in $K$, $H$ also has property $p$ in $K$.

Related notions

Given a subgroup property modifier, we are often interested in the image space of the modifier: the collection of those subgroup properties that can be obtained by applying the modifier to some property. We are also interested in the fixed-point space: the collection of subgroup properties that remain unchanged on applying the modifier.

There are some subgroup property modifiers whose image space is the same as the fixed-point space; this is equivalent to the condition that applying the modifer twice has the same effect as applying it once. Such a subgroup property modifier is termed an idempotent subgroup property modifier.

There are other properties that nice subgroup property modifiers have. Check out: