# 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.

## Related operators

### 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:

### 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