Characteristic subgroup of Sylow subgroup

This page describes a subgroup property obtained as a composition of two fundamental subgroup properties: characteristic subgroup and Sylow subgroup
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Definition

A subgroup of a finite group is termed a characteristic subgroup of Sylow subgroup if it can be expressed as a characteristic subgroup of some Sylow subgroup.

Relation with other properties

Stronger properties

• Sylow subgroup

Weaker properties

• Automorph-conjugate subgroup

Facts

• Frattini's argument, which is generally stated for Sylow subgroups, holds for all automorph-conjugate subgroups. In particular, it holds for characteristic subgroups of Sylow subgroups. In words: if $K\leq L\leq G$ are such that $K$ is a characteristic subgroup of a Sylow subgroup of $L$, and $L$ is normal in $G$, then $G=LN_{G}(K)$.