Subgroup property collapse formalism

Definition
The subgroup property collapse formalism is a technique for expressing a group property in terms of two subgroup properties. Given two subgroup properties $$p$$ and $$q$$, the group property $$p == q$$ is defined as the property of being a group such that the subgroups with property $$p$$ in that are precisely the same as the subgroups with property $$q$$.

Typically, we consider subgroup property collapse formalisms where $$p \le q$$, that is, where a subgroup with property $$p$$ always has property $$q$$ in any group.

The Hamiltonian operator
The Hamiltonian operator on a subgroup property $$q$$ is the group property $$t == q$$, viz the group property that every subgroup in it satisfies property $$q$$ in it.

Examples of the application of the Hamiltonian operator are as follows:


 * Dedekind group or Hamiltonian group: Every subgroup is normal
 * PH-group: Every subgroup is permutable
 * N-group: Every subgroup is ascendant

The transitively operator
The transitively operator on a subgroup property $$q$$ is either of the following:


 * $$q == Sub(q)$$ where $$Sub(q)$$ denotes the subordination of $$q$$
 * $$q == R(q)$$ where $$R(q)$$ denotes the right transiter of $$q$$

Examples of the application of the transitively operator:


 * T-group: The transitively operator applied to the subgroup property of being normal
 * PT-group: The transitively operator applied to the subgroup property of being permutable