# Hall 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 definitionVIEW: Definitions built on this | Facts about this: (factscloselyrelated to Hall subgroup, all facts related to Hall subgroup) |Survey articles about this | Survey articles about definitions built on this

VIEW RELATED: Analogues of this | Variations of this | Opposites of this |

View a complete list of semi-basic definitions on this wiki

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

## Definition

### Definition without prime set specification

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

- The order of is relatively prime to the index of in .
- For any prime number dividing the order of , divides
*exactly one*of the two numbers: the order of and the index of in .

### Definition with prime set specification

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

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

We sometimes use the notation to refer to the complement of in the set of prime numbers.

### Relation between order and prime set specification

- The order of a Hall -subgroup of depends only on the prime set and on the order of . In particular, for fixed , all Hall -subgroups have the same order.
- Conversely, if two Hall subgroups of a group have the same order, then the prime set specifications that work for one Hall subgroup also work for the other.
- As far as the definition of Hall -subgroup of is concerned, we
*only*care about the intersection of with the set of prime divisors of the order of . Adding or removing primes that do not divide the order of does not affect the notion of Hall -subgroup.

## Examples

### Extreme examples

- The trivial subgroup is a Hall subgroup in any finite group. [SHOW MORE]
- Every finite group is a Hall subgroup of itself. [SHOW MORE]

### Sylow subgroups and p-complements

There are two other important near-extremes of Hall subgroups:

- Sylow subgroups are Hall subgroups corresponding to a single prime. In other words, a Sylow subgroup is a finite p-group whose index is relatively prime to . If divides the order of the group, -Sylow subgroups must be nontrivial. Sylow's theorem guarantees the existence and other nice behavior of the -Sylow subgroup for any prime in any finite group.
- p-complements are Hall subgroups whose
*index*is a prime power. In other words, they are Hall subgroups whose prime set excludes at most one prime divisor of the order of the group. A -complement is thus a Hall -subgroup where is the set of primes other than . (As always, we only care about the primes that divide the order of the group).

### Particular examples

- A3 in S3: The subgroup has order 3 and index 2 in a group of order 6. It is a 3-Sylow subgroup and also a 2-complement.
- A4 in A5: The subgroup has order 12 and index 5 in a group of order 60. It is a -Hall subgroup and also a 5-complement.
- S4 in S5: The subgroup has order 24 and index 5 in a group of order 120. It is a -Hall subgroup and also a 5-complement.

Here is a list of examples:

## Facts

### Existence and domination

- [[Existence of pi-subgroups for all prime sets is equivalent to existence of p-complements for all primes p]]
- [[ECD condition for -subgroups in finite solvable groups]]: This states that in finite solvable groups, -Hall subgroup exist for all prime sets , they are conjugate, and they dominate -subgroups.
- Hall's theorem: This is a converse to the above, stating that if -Hall subgroups exist for all prime sets , then the group is solvable.

### Sylow subgroups and other special cases

- Sylow's theorem states for Sylow subgroups (Hall subgroups corresponding to a single prime), the existence, conjugacy, and domination conditions hold in
*all*finite groups, not just in finite solvable groups. - Nilpotent Hall subgroups of same order are conjugate

### General non-existence and other results

- Hall subgroups need not exist
- Hall not implies order-isomorphic: It is possible for two -Hall subgroups of a finite group to be non-isomorphic.
- Hall not implies isomorph-conjugate: It is possible for two isomorphic -Hall subgroups of a finite group to not be conjugate subgroups inside .

## Relation with other properties

### Stronger properties

### Weaker properties

### 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 transitiveABOUT 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 conditionABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

This states that if is a Hall subgroup of and is some subgroup containing , then is a Hall subgroup of .

`For full proof, refer: Hall satisfies intermediate subgroup condition`

### Transfer condition

This subgroup property doesnotsatisfy 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).