Subgroup property modifier

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:

Category:Subgroup property modifier properties