Relation implication expression: Difference between revisions

Latest revision as of 00:07, 8 May 2008

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 $a ⟹ b$ is defined as follows:

$H \leq G$ satisfies $a ⟹ b$ 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 $⟹$ 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 $H$ and $K$ are said to permute if $H K = K H$ or equivalently, if $H K$ 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 $a ⟹$ Permuting.

For instance:

Conjugate-permutable subgroup: Conjugate $⟹$ Permuting
Automorph-permutable subgroup: Automorph $⟹$ Permuting

@@ Line 1: / Line 1: @@
-{{implication formalism for subgroup properties}}
+{{formal expression for subgroup property|Relation-implication-expressible subgroup properties}}
 ==Definition==
@@ Line 5: / Line 5: @@
 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 <math>a</math> and <math>b</math>, the subgroup property <math>a \implies b</math> is defined as follows:
+The relation implication operator takes as input two subgroup relations and outputs a [[subgroup property]], as follows. Given two subgroup relations <math>a</math> and <math>b</math>, the subgroup property <math>a \implies b</math> is defined as follows:
 <math>H \le G</math> satisfies <math>a \implies b</math> if for any subgroup <math>K</math> such that <math>(H,K)</math> satisfies <math>a</math>, <math>(H,K)</math> must also satisfy <math>b</math>.
+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===
@@ Line 25: / Line 31: @@
 Some natural relation implication properties arising from these are:
-* [[Order-unique subgroup]] = Same order &rarr; Same subgroup
+* [[Order-unique subgroup]] = Same order <math>\implies</math> Same subgroup
-* [[Isomorph-free subgroup]] = Isomorphic &rarr; Same subgroup
+* [[Isomorph-free subgroup]] = Isomorphic <math>\implies</math> Same subgroup
-*
+* [[Characteristic subgroup]] = Automorphism <matH>\implies</math>Same subgroup
+* [[Normal subgroup]] = Conjugate <math>\implies</math> Same subgroup
+* [[Order-conjugate subgroup]] = Same order <math>\implies</math> Conjugate subgroups
+* [[Isomorph-conjugate subgroup]] = Isomorph <math>\implies</math> Automorph
+* [[Automorph-conjugate subgroup]] = Automorph <math>\implies</math> Conjugate
+===Permutability===
+Here are some important subgroup relations:
+* [[Permuting subgroups]]: Two subgroups <math>H</math> and <math>K</math> are said to permute if <math>HK=KH</math> or equivalently, if <math>HK</math> is a group.
+* [[Totally permuting subgroups]]: Two subgroups <math>H</math> and <math>K</math> are said to be totally permuting if every subgroup of <math>H</math> permutes with every subgroup of <math>K</math>.
+Given a [[subgroup relation]] <math>a</math>, a subgroup is said to be <math>a</matH>-permutable if it satisfies <math>a \implies</math> Permuting.
+For instance:
+* [[Conjugate-permutable subgroup]]: Conjugate <math>\implies</math> Permuting
+* [[Automorph-permutable subgroup]]: Automorph <math>\implies</math> Permuting