The Group Properties Wiki (pre-alpha)

TIP: Having trouble locating the wiki page about a given fact? Get tips

ABOUT US: We use Semantic MediaWiki. Learn more about using Semantic MediaWiki

ALSO CHECK OUT: Commalg: The Commutative Algebra Wiki

From Groupprops

This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: normal subgroup and Hall subgroup
View other subgroup property conjunctions | view all subgroup properties

This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: subnormal subgroup and Hall subgroup
View other subgroup property conjunctions | view all subgroup properties

This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: characteristic subgroup and Hall subgroup
View other subgroup property conjunctions | view all subgroup properties

Contents 1 Definition 1.1 Symbol-free definition 1.2 Definition with symbols 1.3 Equivalence of definitions 2 Relation with other properties 2.1 Stronger properties 2.2 Weaker properties 3 Metaproperties 3.1 Transitivity 3.2 Transfer condition

Definition

Symbol-free definition

A subgroup of a finite group is said to be a normal Hall subgroup if it satisfies the following equivalent conditions:

1. It is a Hall subgroup, viz., the order and index are relatively prime, and is a normal subgroup viz., every inner automorphism of the whole group takes the subgroup to itself,
2. It is a Hall subgroup and is a characteristic subgroup: every automorphism of the group takes the subgroup to itself.
3. It is a Hall subgroup and is a fully characteristic subgroup: every endomorphism of the group takes the subgroup to itself.
4. It is a Hall subgroup and is a subnormal subgroup.

Definition with symbols

A subgroup H of a group G is said to be a normal Hall subgroup if it satisfies the following equivalent conditions:

1. | H | and [G:H] are relatively prime, and , viz., for any
2. | H | and [G:H] are relatively prime, and , viz., for all
3. | H | and [G:H] are relatively prime, and , viz., for all
4. | H | and [G:H] are relatively prime, and H is subnormal in G.

Equivalence of definitions

Check out: Hall implies intermediately normal-to-characteristic, Hall implies intermediately subnormal-to-normal.

Relation with other properties

Stronger properties

• Hall direct factor
• Normal Sylow subgroup

Weaker properties

• Order-unique subgroup
• Permutably complemented normal subgroup: For full proof, refer: Normal Hall implies permutably complemented
• Permutably complemented subgroup: For full proof, refer: Normal Hall implies permutably complemented
• Homomorph-containing subgroup
• Fully characteristic subgroup: For full proof, refer: Normal Hall implies fully characteristic
• Characteristic subgroup

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

The property of being a normal Hall subgroup is characteristic because:

• The Hall part is transitive
• The normal part becomes transitive because the property of being a Hall subgroup is a transitivizer of normality, or more specifically, because it is a normal-to-characteristic subgroup property, and the subgroup proeprty of being characteristic is transitive.

Transfer condition

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.
View a complete list of such properties

Since both the properties of being normal and of being Hall satisfy the transfer condition, so does the property of being a normal Hall subgroup.