Characteristic subgroup of Sylow subgroup

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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

Weaker properties

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).