Thread: 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|

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|.

I already did this for you earlier in the week...
http://www.mathhelpforum.com/math-he...t-algebra.html