Let and be a finite groups, a Sylow -subgroup of , and . If is normal in and is normal in , prove that is normal in . Deduce that if then . (That is, normalizers of Sylow subgroups are self-normalizing.)
Let . Now since is normal, so that conjugation by is an automorphism of . Since is the unique subgroup of its order in , is characteristic in , and we have . Thus is normal in .
Now let be a Sylow subgroup of . Note that is the -largest subgroup of in which is normal. Now if is normal, we have normal by the above argument, so that . In particular, .