Let be a finite group of order . Let be a subgroup of . Define the normalizer of , and prove that the number of distinct conjugates of is equal to the index of in . If is a prime dividing , deduce that the number of Sylow -subgroups of must divide .
[You may assume the existence and conjugacy of Sylow subgroups.]
Prove that any group of order 72 must have either 1 or 4 Sylow 3-subgroups.
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