Subgroup property collapse formalism
The subgroup property collapse formalism is a technique for expressing a group property in terms of two subgroup properties. Given two subgroup properties and , the group property is defined as the property of being a group such that the subgroups with property in that are precisely the same as the subgroups with property .
Typically, we consider subgroup property collapse formalisms where , that is, where a subgroup with property always has property in any group.
The Hamiltonian operator
The Hamiltonian operator on a subgroup property is the group property , viz the group property that every subgroup in it satisfies property 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 is either of the following:
Examples of the application of the transitively operator: