Let be a group of order where and , , and are primes. Prove that a Sylow -subgroup of is normal.
Recall that some Sylow subgroup of is normal. Let , , and denote Sylow -, -, and -subgroups of , respectively.
Suppose . Note that , and since , . Let (via the lattice isomorphism theorem) be the subgroup whose quotient is the unique Sylow -subgroup in ; we have , and is normal. Moreover, because , has a unique Sylow -subgroup , and we have . Now if is a Sylow -subgroup, then , so that . Since is Sylow, . Thus .
The same argument works if .