Normal subgroup of Sylow subgroup

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Definition

A subgroup of a finite group is termed a normal subgroup of Sylow subgroup if it can be expressed as a normal subgroup of a $p$-Sylow subgroup of the whole for some prime $p$.

Relation with other properties

Stronger properties

• Normal subgroup of prime power order: Any normal $p$-subgroup for a prime $p$ is contained in a $p$-Sylow subgroup, and is also normal in that.
• Sylow subgroup of normal subgroup

Weaker properties

• Subgroup of prime power order