# 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.

## 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 $K \le L \le G$ are such that $K$ is a characteristic subgroup of a Sylow subgroup of $L$, and $L$ is normal in $G$, then $G = LN_G(K)$.