Normalizers an p-Sylow subgroups

[barbutzo]

Homework Statement

Let G be a finite group, and H be a normal subgroup of G. Let B be a p-Sylow subgroup of H, for some p dividing |H|. Show that G=HN_G(B).

(N_G(B) is the normalizer of B in G, that is the biggest subgroup of G which contains B and B is normal in it. Equivalently N_G(B)={g\inG|gBg^-1=B})

Homework Equations

The Sylow theorems.

The Attempt at a Solution

To tell the truth I'm pretty stumped with this question. I know that the index of the normalizer is the number of p-Sylow subgroups, but B is a p-Sylow subgroup of H, and the normalizer is in G. I also know that B must be the only p-Sylow subgroup of its normalizer. I can't see how this adds up to a solution. I also tried considering a left action of H on G, but things seem to abstract to get something from that.

 

[morphism]

This is a classical result, called Frattini's argument.

The proof is pretty straightforward. So if you don't want to look it up, here's a hint: take a g in G and consider g^-1Bg. There are two important things we can say about this - what are they?

 

[barbutzo]

well, first, since H is normal, gBg^-1 is a subgroup of H.
second, it's also a p-Sylow subgroup of H, so it's conjugate to B.

Oh! so it's also of the form hBh^-1!
great, I'll take it from here. thanks!