Composition operator

Definition with symbols
Given two subgroup properties $$p$$ and $$q$$, the composition operator applied to these properties, denoted as $$p * q$$, is the property defined as follows: $$H$$ has the property $$p * q$$ as a subgroup of $$G$$ if there is an intermediate subgroup $$K$$ such that $$H$$ satisfies $$p$$ as a subgroup of $$K$$ and $$K$$ satisfies $$q$$ as a subgroup of $$G$$.

Associativity
Given subgroup properties $$p$$, $$q$$, and $$r$$, the following relation holds:

$$(p * q) * r = p * (q * r)$$

In other words, the composition operator is associative.

Quantalic nature
The composition operator is a monotone operator in both arguments, when the properties are given the usual partial order of implication. Further, it distributes over logical disjunction, and is hence a quantalic property operator.

Identity element
The identity element for the composition operator is the property of being the improper subgroup, that is, of being the group embedded as a subgroup in itself. Any property that is implied by this property is termed an identity-true subgroup property.

Nil element
The nil element for the composition operator is the fallacy subgroup property, that is the subgroup property that is never satisfied.

Transiters
Since the composition operator is an associative quantalic property operator, the Transiter master theorem is applicable to it, and we can talk of left transiters, right transiters. We can also talk of the subordination with respect to this operator.