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Back to our topic on Sylow theory…

Let be a finite group, where is a prime divisor of . Suppose that whenever and are two distinct Sylow q-subgroups of , is a subgroup of of index at least . Prove that the number of Sylow q-subgroups of G satisfies .

Proof: Let be the set of all Sylow q-subgroups of G. Fix . Consider the group action of P acting on by conjugation.

,

By Orbit-Stabilizer Theorem, .

We claim that , since any element x outside of cannot normalise , since otherwise if , , then will be a larger q-subgroup of G than .

Thus, , i.e. .

.

The orbits form a partition of , thus , where the sum runs over all orbits other than .

Thus, .

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