Hall subgroup

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]

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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

• Sylow subgroup
• Sylow complement

Conjunction with other properties

• Normal Hall subgroup

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.

Transfer condition

YES: This subgroup property satisfies the transfer condition: if a subgroup has the property in the whole group, its intersection with any subgroup has the property in that subgroup.
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If $H$ is a Hall subgroup of $G$, and $K$ is any subgroup of $G$, then $H\cap K$ is a Hall subgroup of $K$. This follows from the following two facts:

• The order of $H$ ∩ $K$ divides the order of $H$
• The index of $H$ ∩ $K$ in $K$ divides the index of $H$ in $G$

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 <amth>G</math> and $K$ is some subgroup containing $H$, then $H$ is a Hall subgroup of $K$.