Sylow retract
From Groupprops
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
Contents
Definition
Symbol-free definition
A Sylow retract of a finite group is any of the following equivalent things:
- A Sylow subgroup that is also a retract. In other words, it is a
-Sylow subgroup for some prime
, such that there exists a normal p-complement: a normal Hall subgroup that is a permutable complement to it.
- 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