I know that, given a finite group group $G$ with Sylow $p$ subgroup $S$, $N_G(S)=N_G(N_G(S))$. I cannot think of how I would find a subgroup $H$ such that $N_G(H)$ is a proper subset of $N_G(N_G(H))$. What is a way to think of such an example?
2026-03-28 12:57:03.1774702623
Normalizer of Normalizer
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A nilpotent group fulfills the normalizer condition: any proper subgroup is always strictly contained in its normalizer, Well, now just choose a non-normal subgroup there...
For an easy example, choose a finite non-abelian $\;p\,-$ group which is not the quaternions (since any subgroup here is normal...).