Hall subgroup

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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

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.
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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