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: 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 without prime set specification
- 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
- 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.
- 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).
- 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:
|Group part||Subgroup part||Quotient part|
|A3 in S3||Symmetric group:S3||Cyclic group:Z3||Cyclic group:Z2|
|A4 in A5||Alternating group:A5||Alternating group:A4|
|S2 in S3||Symmetric group:S3||Cyclic group:Z2|
|S4 in S5||Symmetric group:S5||Symmetric group:S4|
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
Relation with other properties
Conjunction with other properties
- Normal Hall subgroup: These are fully characteristic. Thus, this subgroup property is normal-to-characteristic
- 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
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
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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 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
This subgroup property does not satisfy the transfer condition
For full proof, refer: Hall does not satisfy transfer condition
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).