# Math Help - Normalizer of a Group

1. ## Normalizer of a Group

Let G be a finite group and let H be a subgroup of G.
NH, the normalizer of H in G is defined to be the set {x|xHx^-1}

Prove that NH is a subgroup of G

Prove that |{xHx^-1|x in G}| = |G/NH|

2. Originally Posted by Coda202
Let G be a finite group and let H be a subgroup of G.
NH, the normalizer of H in G is defined to be the set {x|xHx^-1}

Prove that NH is a subgroup of G

Prove that |{xHx^-1|x in G}| = |G/NH|
Define the action of G on its subgroups by conjugation.
Then the orbit of H is {xHx^{-1} | x in G} and the stabilizer of H is NH.
Now it follows by orbit-stabilizer theorem that |{xHx^{-1}|x in G}| = |G/NH|.