(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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})

2. Relevant equations

The Sylow theorems.

3. 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.

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# Normalizers an p-Sylow subgroups

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