Relation implication expression

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:


 * 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