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
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