Hall subgroup: Difference between revisions

Revision as of 08:53, 22 February 2007

Origin

The notion of Hall subgroup was introduced by [[Philip Hall] who studied their properties and proved the theorem that a group is solvable if and only if it has Hall subgroups of all possible orders.

Definition

Symbol-free definition

A subgroup of a finite group is termed a Hall subgroup if its order and index are coprime.

We also have a notion of Hall subgroup in a profinite group which generalizes the above notion of Hall subgroup.

Definition with symbols

A subgroup $H$ of a finite group $G$ is termed a Hall subgroup if the order of H (viz the cardinality of $H$ as a set) is coprime to the index of $H$ (viz the number of cosets of $H$ in $G$ ).

Equivalently, $H$ is a Hall subgroup if for any prime dividing the order of $G$ , either the prime is fully inside the order of $H$ or fully inside the index of $H$ .

Relation with other properties

Stronger properties

Conjunction with other properties

Normal Hall subgroup

Metaproperties

Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.
ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity

A Hall subgroup of a Hall subgroup is a Hall subgroup. This follows from the fact that the index is multiplicative. For full proof, refer: Hall satisfies transitivity

Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties

The property of being a Hall subgroup is trivially true, that is, the trivial subgroup is a Hall subgroup in any group.

It is also identity-true, that is, every finite group is a Hall subgroup of itself.

Transfer condition

YES: This subgroup property satisfies the transfer condition: if a subgroup has the property in the whole group, its intersection with any subgroup has the property in that subgroup.
View other subgroup properties satisfying the transfer condition

If $H$ is a Hall subgroup of $G$ , and $K$ is any subgroup of $G$ , then $H \cap K$ is a Hall subgroup of $K$ . This follows from the following two facts:

The order of $H$ ∩ $K$ divides the order of $H$
The index of $H$ ∩ $K$ in $K$ divides the index of $H$ in $G$

Intermediate subgroup condition

YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

This states that if $H$ is a Hall subgroup of <amth>G</math> and $K$ is some subgroup containing $H$ , then $H$ is a Hall subgroup of $K$ .

@@ Line 1: / Line 1: @@
 ==Origin==
-The notion of Hall subgroup, who studied their properties and proved the theorem that a group is solvable if and only if it has Hall subgroups of all possible orders.
+The notion of Hall subgroup was introduced by [[Philip Hall] who studied their properties and proved the theorem that a group is solvable if and only if it has Hall subgroups of all possible orders.
 ==Definition==
@@ Line 30: / Line 30: @@
 ==Metaproperties==
-===Transitivity===
+{{transitive}}
 A Hall subgroup of a Hall subgroup is a Hall subgroup. This follows from the fact that the [[index of a subgroup#multiplicativity|index is multiplicative]]. {{proofat|[[Hall satisfies transitivity]]}}
-===Trimness===
+{{trim}}
 The property of being a Hall subgroup is [[trivially true subgroup property|trivially true]], that is, the trivial subgroup is a Hall subgroup in any group.
@@ Line 40: / Line 40: @@
 It is also [[identity-true subgroup property|identity-true]], that is, every finite group is a Hall subgroup of itself.
-===Intermediate subgroup condition===
+{{transfercondn}}
-A Hall subgroup of a group is also a Hall subgroup in every intermediate subgroup. The property of being a Hall subgroup also satisfies the [[transfer condition]].
+If <math>H</math> is a Hall subgroup of <math>G</math>, and <math>K</math> is any subgroup of <math>G</math>, then <math>H \cap K</math> is a Hall subgroup of <math>K</math>. This follows from the following two facts:
-[[Category: Finite subgroup properties]]
+* The [[order of a group|order]] of <math>H</math> &cap; <math>K</math> divides the order of <math>H</math>
-[[Category: Order-determined finite subgroup properties]]
+* The [[index of a subgroup|index]] of <math>H</math> &cap; <math>K</math> in <math>K</math> divides the index of <math>H</math> in <math>G</math>
-[[Category: Transitive subgroup properties]]
-[[Category: Subgroup properties]]
+{{intsubcondn}}
-[[Category: Trim subgroup properties]]
-[[Category: Subgroup properties satisfying intermediate subgroup condition]]
+This states that if <math>H</math> is a Hall subgroup of <amth>G</math> and <math>K</math> is some subgroup containing <math>H</math>, then <math>H</math> is a Hall subgroup of <math>K</math>.