On the Frattini lemma

TL;DR

This paper proves that if a subgroup K of a finite group G satisfies a certain Sylow subgroup condition involving normalizers, then K must be a normal subgroup of G.

Contribution

The paper establishes a new criterion for subgroup normality based on Sylow subgroup normalizers in finite groups.

Findings

K is normal in G under the given conditions

Provides a new perspective on subgroup normality criteria

Enhances understanding of subgroup structure in finite groups

Abstract

Let $K$ be a subgroup of a finite group $G$, and suppose that $G=KN_G(P)$ for every Sylow subgroup $P$ of $K$. Then the subgroup $K$ is normal in $G$.