Hall direct factor

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

Definition

A subgroup of a finite group is termed a Hall direct factor if it satisfies the following equivalent conditions:

1. It is a Hall subgroup and is also a direct factor.
2. It is a normal Hall subgroup and possesses a normal complement, i.e., is a retract.
3. It is a Hall subgroup and is also a central factor of the whole group.

Equivalence of definitions

For full proof, refer: Hall and central factor implies direct factor

Relation with other properties

Stronger properties

• Sylow direct factor

Weaker properties

• Normal Hall subgroup
• Fully characteristic subgroup
• Fully characteristic direct factor
• Direct factor
• Central factor
• Conjugacy-closed normal subgroup
• Conjugacy-closed Hall 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

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 other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties

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.
ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition