I Notation N(H) for a subgroup

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1. Oct 28, 2016

Silviu

Hello! I have this problem:

If H is a subgroup of prime index in a finite group G, show that either H is a normal subgroup or N(H) = H.

What does N(H) means? I don't want a solution for the problem (at least not yet), I just want to know what that notation means. Thank you!

2. Oct 28, 2016

Staff: Mentor

N(H) or better $N_G(H)$ is very likely the normalizer of $H$ in $G$.
That is $N_G(H)=\{g \in G \,\vert \, gHg^{-1} \subseteq H\}\,$.
The task here is to show that either $N_G(H)=G$ or $N_G(H)=H\,$.
The first means a normal subgroup, the latter is called a self-normalizing subgroup.