Sylow direct factor

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

Definition

Symbol-free definition

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

  1. It is a Sylow subgroup and is also a direct factor of the group.
  2. It is a normal Sylow subgroup and possesses a normal p-complement, i.e., it is a retract of the group.
  3. It is a Sylow subgroup and is also a central factor of the whole group.
  4. It is a Sylow subgroup and is also a conjugacy-closed normal subgroup of the whole group.

Equivalence of definitions

Further information: Equivalence of definitions of Sylow direct factor

Relation with other properties

Weaker properties

Metaproperties

Join-closedness

YES: This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property.
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ABOUT JOIN-CLOSEDNESS: View all join-closed subgroup properties (or, strongly join-closed properties) | View all subgroup properties that are not join-closed | Read a survey article on proving join-closedness | Read a survey article on disproving join-closedness

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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ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition