Wilson’s theorem is a useful theorem in Number Theory, and may be proved in several different ways. One of the interesting proofs is to prove it using Sylow’s Third Theorem.

Let , the symmetric group on p elements, where p is a prime.

By Sylow’s Third Theorem, we have . The Sylow p-subgroups of have p-cycles each.

There are a total of different p-cycles (cyclic permutations of p elements).

Thus, we have , which implies that

Thus , and multiplying by p-1 gives us which is precisely Wilson’s Theorem. 🙂

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