Sylow retract

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 defining ingredient::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.

Stronger properties

 * Weaker than::Sylow direct factor

Weaker properties

 * Stronger than::Hall retract
 * Stronger than::Retract
 * Stronger than::Conjugacy-closed subgroup