Hall subgroup

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

All the primes dividing the order of $H$ are in the prime set $\pi$ and all the primes dividing the index of $H$ in $G$ are outside the prime set $\pi$ .
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 $\pi$ . In other words, it is the $\pi$ -part of the order of $G$ .

We sometimes use the notation $\pi '$ to refer to the complement of $\pi$ in the set of prime numbers.

Relation between order and prime set specification

The order of a Hall $\pi$ -subgroup of $G$ depends only on the prime set $\pi$ and on the order of $G$ . In particular, for fixed $\pi$ , all Hall $\pi$ -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 $\pi$ -subgroup of $G$ is concerned, we only care about the intersection of $\pi$ 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 $\pi$ -subgroup.

Examples

Extreme examples

The trivial subgroup is a Hall subgroup in any finite group. [SHOW MORE]
In terms of prime set specifications, it is the Hall subgroup corresponding to the empty set of primes. Equivalently it is the Hall subgroup corresponding to any subset of the set of all primes that does not intersect the set of primes dividing the order of the group.
Every finite group is a Hall subgroup of itself. [SHOW MORE]
In terms of prime set specifications, it is the Hall subgroup corresponding to all primes dividing the order of the group. Equivalently, it is the Hall subgroup corresponding to any subset of the set of all primes that contains all prime divisors of the order of the group.

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 $p$ . If $p$ divides the order of the group, $p$ -Sylow subgroups must be nontrivial. Sylow's theorem guarantees the existence and other nice behavior of the $p$ -Sylow subgroup for any prime $p$ 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 $p$ -complement is thus a Hall $p'$ -subgroup where $p'$ is the set of primes other than $p$ . (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 $\{2,3\}$ -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 $\{2,3\}$ -Hall subgroup and also a 5-complement.

Here is a list of examples:

	Group part	Subgroup part	Quotient part
S3 in S5	Symmetric group:S5	Symmetric group:S3
S4 in S5	Symmetric group:S5	Symmetric group:S4

Facts

Existence and domination

[[Existence of pi-subgroups for all prime sets $\pi$ is equivalent to existence of p-complements for all primes p]]
[[ECD condition for $\pi$ -subgroups in finite solvable groups]]: This states that in finite solvable groups, $\pi$ -Hall subgroup exist for all prime sets $\pi$ , they are conjugate, and they dominate $\pi$ -subgroups.
Hall's theorem: This is a converse to the above, stating that if $\pi$ -Hall subgroups exist for all prime sets $\pi$ , 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 $\pi$ -Hall subgroups of a finite group $G$ to be non-isomorphic.
Hall not implies isomorph-conjugate: It is possible for two isomorphic $\pi$ -Hall subgroups of a finite group $G$ to not be conjugate subgroups inside $G$ .

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

Property	Meaning	Proof of implication	Intermediate notions
join of Sylow subgroups	join of Sylow subgroups of the whole group	Hall implies join of Sylow subgroups	\|FULL LIST, MORE INFO
join of automorph-conjugate subgroups	join of automorph-conjugate subgroups of the whole group	(via join of Sylow sugroups)	\|FULL LIST, MORE INFO
core-characteristic subgroup	normal core is a characteristic subgroup		\|FULL LIST, MORE INFO
closure-characteristic subgroup	normal closure is a characteristic subgroup		\|FULL LIST, MORE INFO
paracharacteristic subgroup	contranormal in its join with any automorphic subgroup	(via join of Sylow subgroups, see also Hall implies paracharacteristic)	\|FULL LIST, MORE INFO
paranormal subgroup	contranormal in its join with any conjugate subgroup	(via paracharacteristic)	\|FULL LIST, MORE INFO
polycharacteristic subgroup		(via paracharacteristic)	\|FULL LIST, MORE INFO
polynormal subgroup		(via paranoral, also via polycharacteristic)	\|FULL LIST, MORE INFO
intermediately normal-to-characteristic subgroup	for any intermediate subgroup in which it is normal, it is also characteristic in that subgroup		\|FULL LIST, MORE INFO
intermediately subnormal-to-normal subgroup	for any intermediate subgroup in which it is subnormal, it is also normal in that subgroup		\|FULL LIST, MORE INFO

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.
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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).
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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.
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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).