The normalizer of a Sylow p-subgroup is “self-normalizing”, i.e. its normalizer is itself. Something that is quite cool.
If is a Sylow -subgroup of a finite group , then .
(Adapted from Hungerford pg 95)
Let . Let , so that . Then is a Sylow -subgroup of . Since is normal in , is the only Sylow -subgroup of . Therefore . This implies . We have proved .
Let Then certainly , so that . Thus .