Sylow retract

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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 retract
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 conjugacy-closed subgroup
View other subgroup property conjunctions | view all subgroup properties

Definition

Symbol-free definition

A Sylow retract of a finite group is any of the following equivalent things:

  1. A Sylow subgroup that is also a retract. In other words, it is a p-Sylow subgroup for some prime p, such that there exists a normal p-complement: a normal Hall subgroup that is a permutable complement to it.
  2. A Sylow subgroup that is also conjugacy-closed: In other words, it is a Sylow subgroup with the property that any two elements of the subgroup that are conjugate in the whole group are conjugate in the subgroup.

Equivalence of definitions

Further information: Conjugacy-closed and Sylow implies retract

Relation with other properties

Stronger properties

Weaker properties