Relation implication expression

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This page describes a formal expression, or formalism, that can be used to describe certain subgroup properties.


View a complete list of formal expressions for subgroup properties OR View subgroup properties expressible using this formalism

Definition

A subgroup relation is a property that can be evaluated for an ordered pair of subgroups of a group. It can thus be thought of as a property over ordered pairs of subgroups in the same group.

The relation implication operator takes as input two subgroup relations and outputs a subgroup property, as follows. Given two subgroup relations a and b, the subgroup property a \implies b is defined as follows:

H \le G satisfies a \implies b if for any subgroup K such that (H,K) satisfies a, (H,K) must also satisfy b.

An expression of a subgroup property in terms of a relation implication operator between subgroup relations, is termed a relation implication expression.

Examples

Note that technically, every subgroup property can be expressed via a relation implication. However, it is not true that every subgroup property benefits from being viewed using a relation implication expression. For a complete list of subgroup properties for which such an expression is useful, refer:

Category:Relation-implication-expressible subgroup properties

Equivalence relation implications

Some important equivalence relations are:

  • Having the same order
  • Being isomorphic as abstract groups
  • Being automorphs, that is, being subgroups such that one can be taken to the other via an automorphism of the whole group
  • Being conjugate subgroups, that is, being subgroups such that one can be taken to the other via an inner automorpism of the whole group
  • Being the same subgroup

These equivalence relations are in increasing order of fineness.

Some natural relation implication properties arising from these are:

Permutability

Here are some important subgroup relations:

Given a subgroup relation a, a subgroup is said to be a-permutable if it satisfies a \implies Permuting.

For instance: