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).

(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)).


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