Sylow subgroup of normal subgroup

This page describes a subgroup property obtained as a composition of two fundamental subgroup properties: Sylow subgroup and normal subgroup
View other such compositions|View all subgroup properties

Definition

Symbol-free definition

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

1. It is a Sylow subgroup of a normal subgroup of the whole group.
2. It is the intersection of a normal subgroup of the whole group with a Sylow subgroup of the whole group.
3. It is a Sylow subgroup inside its normal closure.

Equivalence of definitions

Further information: Equivalence of definitions of Sylow subgroup of normal subgroup

Relation with other properties

Stronger properties

• Sylow subgroup

Weaker properties

• Intermediately isomorph-conjugate subgroup of normal subgroup
• Pronormal subgroup: For full proof, refer: Sylow of normal implies pronormal
• Normal subgroup of Sylow subgroup
• Sylow subgroup of permutable subgroup