Normal Hall subgroup

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

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 $H \triangleleft G$, viz., $gHg^{-1} \leq H$ for any $g \in G$
2. $|H|$ and $[G:H]$ are relatively prime, and $H \operatorname{char} G$, viz., $\sigma(H) \le H$ for all $\sigma \in \operatorname{Aut}(G)$
3. $|H|$ and $[G:H]$ are relatively prime, and $H \operatorname{char} G$, viz., $\sigma(H) \le H$ for all $\sigma \in \operatorname{End}(G)$
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 in the whole group.
ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity

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

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.
View other subgroup properties satisfying the transfer condition

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.