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From Groupprops

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof.
View a complete list of subgroup properties|Get subgroup property lookup help |Get exploration suggestions
VIEW RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions | Subgroup metaproperty dissatisfactions | |
RANDOM SUBGROUP PROPERTY: 2-subnormal subgroup: A normal subgroup of a normal subgroup. Need not be normal, because normality is not transitive.

Contents 1 Origin 2 Definition 2.1 Symbol-free definition 2.2 Definition with symbols 3 Relation with other properties 3.1 Stronger properties 3.2 Weaker properties 3.3 Conjunction with other properties 3.4 Incomparable properties 4 Metaproperties 4.1 Transitivity 4.2 Trimness 4.3 Intermediate subgroup condition 4.4 Transfer condition

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.
View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties

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)
View all trim subgroup properties OR view trivially true subgroup properties OR view 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

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
View all subgroup properties satisfying the intermediate subgroup condition|View facts related to the intermediate subgroup condition

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