Hall subgroup

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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]

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This article defines a subgroup property that makes sense within a finite group

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
• Normal Hall subgroup
• Normal Sylow subgroup
• Hall retract
• Sylow retract
• Nilpotent Hall subgroup
• Order-dominating Hall subgroup
• Order-conjugate Hall subgroup
• Isomorph-conjugate Hall subgroup
• Pronormal Hall subgroup

Weaker properties

• Join of Sylow subgroups: For full proof, refer: Hall implies join of Sylow subgroups
• Join of automorph-conjugate subgroups
• Core-characteristic subgroup
• Closure-characteristic subgroup
• Paracharacteristic subgroup: For full proof, refer: Hall implies paracharacteristic
• Paranormal subgroup: For full proof, refer: Hall implies paranormal
• Polycharacteristic subgroup
• Polynormal subgroup
• Intermediately normal-to-characteristic subgroup
• Intermediately subnormal-to-normal subgroup

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