Sylow retract

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

• Sylow direct factor

Weaker properties

• Hall retract
• Retract
• Conjugacy-closed subgroup