Subgroup property collapse formalism

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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:

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