# Subgroup property collapse formalism

## Contents

## Definition

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.

## Related operators

### 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:

- where denotes the subordination of
- where denotes the right transiter of

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