# Relation implication expression

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

thisformalism

## 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 and , the subgroup property is defined as follows:

satisfies if for any subgroup such that satisfies , must also satisfy .

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:

- Order-unique subgroup = Same order Same subgroup
- Isomorph-free subgroup = Isomorphic Same subgroup
- Characteristic subgroup = Automorphism Same subgroup
- Normal subgroup = Conjugate Same subgroup
- Order-conjugate subgroup = Same order Conjugate subgroups
- Isomorph-conjugate subgroup = Isomorph Automorph
- Automorph-conjugate subgroup = Automorph Conjugate

### Permutability

Here are some important subgroup relations:

- Permuting subgroups: Two subgroups and are said to permute if or equivalently, if is a group.
- Totally permuting subgroups: Two subgroups and are said to be totally permuting if every subgroup of permutes with every subgroup of .

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

For instance:

- Conjugate-permutable subgroup: Conjugate Permuting
- Automorph-permutable subgroup: Automorph Permuting