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 this formalism
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.
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:
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
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.