Hall subgroup: Difference between revisions

Revision as of 00:19, 23 March 2012

Definition

Definition without prime set specification

A subgroup $H$ of a finite group $G$ is termed a Hall subgroup if it satisfies the following equivalent conditions:

The order of $H$ is relatively prime to the index of $H$ in $G$ .
For any prime number $p$ dividing the order of $G$ , $p$ divides exactly one of the two numbers: the order of $H$ and the index of $H$ in $G$ .

Definition with prime set specification

Suppose $π$ is a set of prime numbers and $G$ is a finite group. A subgroup $H$ of $G$ is termed a $π$ -Hall subgroup or Hall $π$ -subgroup if it satisfies the following equivalent conditions:

All the primes dividing the order of $H$ are in the prime set $π$ and all the primes dividing the index of $H$ in $G$ are outside the prime set $π$ .
The order of $G$ is the unique largest divisor of the order of $G$ that has the property that all its prime divisors are in $π$ . In other words, it is the $π$ -part of the order of $G$ .

We sometimes use the notation $π^{'}$ to refer to the complement of $π$ in the set of prime numbers.

Note a few things regarding this definition:

The order of a Hall $π$ -subgroup of $G$ depends only on the prime set $π$ and on the order of $G$ . In particular, for fixed $π$ , all Hall $π$ -subgroups have the same order.
As far as the definition of Hall $π$ -subgroup of $G$ is concerned, we only care about the intersection of $π$ with the set of prime divisors of the order of $G$ . Adding or removing primes that do not divide the order of $G$ does not affect the notion of Hall $π$ -subgroup.

This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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This article defines a subgroup property that makes sense within a finite group

Relation with other properties

Stronger properties

Property	Meaning	Proof of strictness (reverse implication failure)	Intermediate notions
Sylow subgroup	Hall subgroup for a single prime, i.e., finite p-group whose index is relatively prime to $p$ for some prime $p$		\|FULL LIST, MORE INFO
p-complement	Hall $p^{'}$ -subgroup, i.e., Hall subgroup whose index is a prime power		\|FULL LIST, MORE INFO
normal Hall subgroup	Hall subgroup that is also a normal subgroup		\|FULL LIST, MORE INFO
Normal Sylow subgroup	Sylow subgroup that is also a normal subgroup		\|FULL LIST, MORE INFO
Hall retract	Hall subgroup that is also a retract, i.e., it has a normal complement. Note that the normal complement must also be a Hall subgroup for the complementary set of primes		\|FULL LIST, MORE INFO
Sylow retract	Sylow subgroup that is also a retract, i.e., $p$ -Sylow subgroup in a group that has a normal p-complement		\|FULL LIST, MORE INFO
nilpotent Hall subgroup	Hall subgroup that is also a nilpotent group		\|FULL LIST, MORE INFO
order-dominating Hall subgroup	Hall subgroup that is also an order-dominating subgroup, i.e., any subgroup of the whole group whose order divides it is conjugate to a subgroup of it	Hall not implies order-dominating	\|FULL LIST, MORE INFO
order-conjugate Hall subgroup	Hall subgroup that is also an order-conjugate subgroup, i.e., all Hall subgroups of that order are conjugate subgroups	Hall not implies order-conjugate	\|FULL LIST, MORE INFO
isomorph-conjugate Hall subgroup	Hall subgroup that is also an isomorph-conjugate subgroup, i.e., it is conjugate to all isomorphic subgroups	Hall not implies isomorph-conjugate	\|FULL LIST, MORE INFO
pronormal Hall subgroup	Hall subgroup that is also a pronormal subgroup		\|FULL LIST, MORE INFO

Weaker properties

Join of Sylow subgroups: For full proof, refer: Hall implies join of Sylow subgroups
Join of automorph-conjugate subgroups
Core-characteristic subgroup
Closure-characteristic subgroup
Paracharacteristic subgroup: For full proof, refer: Hall implies paracharacteristic
Paranormal subgroup: For full proof, refer: Hall implies paranormal
Polycharacteristic subgroup
Polynormal subgroup
Intermediately normal-to-characteristic subgroup
Intermediately subnormal-to-normal subgroup

Conjunction with other properties

Normal Hall subgroup: These are fully characteristic. Thus, this subgroup property is normal-to-characteristic

Incomparable properties

Order-isomorphic subgroup: Two Hall subgroups of the same order need not be isomorphic. For full proof, refer: Hall not implies order-isomorphic
Isomorph-automorphic subgroup: Two isomorphic Hall subgroups of the same order need not be automorphs. For full proof, refer: Hall not implies isomorph-automorphic
Automorph-conjugate subgroup: Two Hall subgroups that are automorphs of each other, need not be conjugate. For full proof, refer: Hall not implies automorph-conjugate

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.

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 $G$ and $K$ is some subgroup containing $H$ , then $H$ is a Hall subgroup of $K$ .

For full proof, refer: Hall satisfies intermediate subgroup condition

Transfer condition

This subgroup property does not satisfy the transfer condition

For full proof, refer: Hall does not satisfy transfer condition

History

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. (see ECD condition for pi-subgroups in finite solvable groups and Hall's theorem).

@@ Line 1: / Line 1: @@
-{{semibasicdef}}
+==Definition==
-{{subgroup property}}
-{{subgroup property (finite groups)}}
+===Definition without prime set specification===
-[[importance rank::2| ]]
-==Origin==
+A [[subgroup]] <math>H</math> of a [[finite group]] <math>G</math> is termed a '''Hall subgroup''' if it satisfies the following equivalent conditions:
-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.
+* The [[order of a group|order]] of <math>H</math> is relatively prime to the [[index of a subgroup|index]] of <matH>H</math> in <matH>G</math>.
+* For any prime number <math>p</math> dividing the order of <math>G</math>, <math>p</math> divides ''exactly one'' of the two numbers: the order of <math>H</math> and the index of <math>H</math> in <math>G</math>.
-==Definition==
+===Definition with prime set specification===
-===Symbol-free definition===
+Suppose <math>\pi</math> is a set of [[prime number]]s and <math>G</math> is a [[finite group]]. A [[subgroup]] <math>H</math> of <math>G</math> is termed a <math>\pi</math>-'''Hall subgroup''' or '''Hall <math>\pi</math>-subgroup''' if it satisfies the following equivalent conditions:
-A subgroup of a finite group is termed a '''Hall subgroup''' if its [[order of a group|order]] and [[index of a subgroup|index]] are coprime.
+# All the primes dividing the [[order of a group|order]] of <math>H</math> are in the prime set <math>\pi</math> and all the primes dividing the [[index of a subgroup|index]] of <math>H</math> in <math>G</math> are outside the prime set <math>\pi</math>.
+# The [[order of a group|order]] of <math>G</math> is the unique largest divisor of the order of <math>G</math> that has the property that all its prime divisors are in <math>\pi</math>. In other words, it is the <math>\pi</math>-part of the order of <math>G</math>.
-We also have a notion of [[Hall subgroup (profinite groups)|Hall subgroup in a profinite group]] which generalizes the above notion of Hall subgroup.
+We sometimes use the notation <math>\pi'</math> to refer to the complement of <math>\pi</math> in the set of prime numbers.
-===Definition with symbols===
+Note a few things regarding this definition:
-A subgroup <math>H</math> of a finite group <math>G</math> is termed a Hall subgroup if the order of H (viz the cardinality of <math>H</math> as a set) is coprime to the index of <math>H</math> (viz the number of cosets of <math>H</math> in <math>G</math>).
+* The order of a Hall <math>\pi</math>-subgroup of <math>G</math> depends only on the prime set <math>\pi</math> and on the order of <math>G</math>. In particular, for fixed <matH>\pi</math>, all Hall <math>\pi</math>-subgroups have the same order.
+* As far as the definition of Hall <math>\pi</matH>-subgroup of <math>G</matH> is concerned, we ''only'' care about the intersection of <matH>\pi</math> with the set of prime divisors of the order of <math>G</math>. Adding or removing primes that do not divide the order of <math>G</math> does not affect the notion of Hall <math>\pi</math>-subgroup.
-Equivalently, <math>H</math> is a Hall subgroup if for any prime dividing the order of <math>G</math>, either the prime is ''fully'' inside the order of <math>H</math> or ''fully'' inside the index of <math>H</math>.
+{{semibasicdef}}
+{{subgroup property}}
+{{subgroup property (finite groups)}}
+[[importance rank::2| ]]
 ==Relation with other properties==
@@ Line 25: / Line 31: @@
 ===Stronger properties===
-* [[Weaker than::Sylow subgroup]]
+{| class="sortable" border="1"
-* [[Weaker than::Sylow complement]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[Weaker than::Normal Hall subgroup]]
+|-
-* [[Weaker than::Normal Sylow subgroup]]
+| [[Weaker than::Sylow subgroup]] || Hall subgroup for a single prime, i.e., [[finite p-group]] whose index is relatively prime to <math>p</math> for some prime <math>p</math> || || || {{intermediate notions short|Hall subgroup|Sylow subgroup}}
-* [[Weaker than::Hall retract]]
+|-
-* [[Weaker than::Sylow retract]]
+| [[Weaker than::p-complement]] || Hall <math>p'</math>-subgroup, i.e., Hall subgroup whose index is a prime power || || || {{intermediate notions short|Hall subgroup|p-complement}}
-* [[Weaker than::Nilpotent Hall subgroup]]
+|-
-* [[Weaker than::Order-dominating Hall subgroup]]
+| [[Weaker than::normal Hall subgroup]] || Hall subgroup that is also a [[normal subgroup]] || || || {{intermediate notions short|Hall subgroup|normal Hall subgroup}}
-* [[Weaker than::Order-conjugate Hall subgroup]]
+|-
-* [[Weaker than::Isomorph-conjugate Hall subgroup]]
+| [[Weaker than::Normal Sylow subgroup]] || Sylow subgroup that is also a [[normal subgroup]] || || || {{intermediate notions short|Hall subgroup|normal Sylow subgroup}}
-* [[Weaker than::Pronormal Hall subgroup]]
+|-
+| [[Weaker than::Hall retract]] || Hall subgroup that is also a [[retract]], i.e., it has a [[normal complement]]. Note that the normal complement must also be a [[Hall subgroup]] for the complementary set of primes || || || {{intermediate notions short|Hall subgroup|Hall retract}}
+|-
+| [[Weaker than::Sylow retract]] || Sylow subgroup that is also a retract, i.e., <math>p</math>-Sylow subgroup in a group that has a [[normal p-complement]] || || || {{intermediate notions short|Hall subgroup|Sylow retract}}
+|-
+| [[Weaker than::nilpotent Hall subgroup]] || Hall subgroup that is also a [[nilpotent group]] || || || {{intermediate notions short|Hall subgroup|nilpotent Hall subgroup}}
+|-
+| [[Weaker than::order-dominating Hall subgroup]] || Hall subgroup that is also an [[order-dominating subgroup]], i.e., any subgroup of the whole group whose order divides it is conjugate to a subgroup of it || || [[Hall not implies order-dominating]] || {{intermediate notions short|Hall subgroup|order-dominating Hall subgroup}}
+|-
+| [[Weaker than::order-conjugate Hall subgroup]] || Hall subgroup that is also an [[order-conjugate subgroup]], i.e., all Hall subgroups of that order are [[conjugate subgroups]] || || [[Hall not implies order-conjugate]] || {{intermediate notions short|Hall subgroup|order-conjugate Hall subgroup}}
+|-
+| [[Weaker than::isomorph-conjugate Hall subgroup]] || Hall subgroup that is also an [[isomorph-conjugate subgroup]], i.e., it is [[conjugate subgroups|conjugate]] to all isomorphic subgroups || || [[Hall not implies isomorph-conjugate]] || {{intermediate notions short|Hall subgroup|isomorph-conjugate Hall subgroup}}
+|-
+| [[Weaker than::pronormal Hall subgroup]] || Hall subgroup that is also a [[pronormal subgroup]] || || || {{intermediate notions short|Hall subgroup|pronormal Hall subgroup}}
+|}
 ===Weaker properties===
@@ Line 81: / Line 101: @@
 {{proofat|[[Hall does not satisfy transfer condition]]}}
+==History==
+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. (see [[ECD condition for pi-subgroups in finite solvable groups]] and [[Hall's theorem]]).