Normalizer of Normalizer of Sylow p-subgroup

The normalizer of a Sylow p-subgroup is “self-normalizing”, i.e. its normalizer is itself. Something that is quite cool.

If $P$ is a Sylow $p$ -subgroup of a finite group $G$ , then $N_G(N_G(P))=N_G(P)$ .

Proof
(Adapted from Hungerford pg 95)

Let $N=N_G(P)$ . Let $x\in N_G(N)$ , so that $xNx^{-1}=N$ . Then $xPx^{-1}$ is a Sylow $p$ -subgroup of $N\leq G$ . Since $P$ is normal in $N$ , $P$ is the only Sylow $p$ -subgroup of $N$ . Therefore $xPx^{-1}=P$ . This implies $x\in N$ . We have proved $N_G(N_G(P))\subseteq N_G(P)$ .

Let $y\in N_G(P)$ Then certainly $yN_G(P)y^{-1}=N_G(P)$ , so that $y\in N_G(N_G(P))$ . Thus $N_G(P)\subseteq N_G(N_G(P))$ .

Author: mathtuition88

Math and Education Blog View all posts by mathtuition88

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