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 $a$ 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:

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 $\implies$ Same subgroup
• Isomorph-free subgroup = Isomorphic $\implies$ Same subgroup
• Characteristic subgroup = Automorphism $\implies$Same subgroup
• Normal subgroup = Conjugate $\implies$ Same subgroup
• Order-conjugate subgroup = Same order $\implies$ Conjugate subgroups
• Isomorph-conjugate subgroup = Isomorph $\implies$ Automorph
• Automorph-conjugate subgroup = Automorph $\implies$ Conjugate

Permutability

Here are some important subgroup relations:

• Permuting subgroups: Two subgroups $H$ and $K$ are said to permute if $HK=KH$ or equivalently, if $HK$ is a group.
• Totally permuting subgroups: Two subgroups $H$ and $K$ are said to be totally permuting if every subgroup of $H$ permutes with every subgroup of $K$.

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

For instance:

• Conjugate-permutable subgroup: Conjugate $\implies$ Permuting
• Automorph-permutable subgroup: Automorph $\implies$ Permuting