Wilson's theorem
Proof using the Sylow theorems

It is possible to deduce Wilson's theorem from a particular application of the Sylow theorems. Let p be a prime. It is immediate to deduce that the symmetric group S p {\displaystyle S_{p}} has exactly (p − 1) ! {\displaystyle (p-1)!} elements of order p, namely the p -cycles C p {\displaystyle C_{p}}. On the other hand, each Sylow p -subgroup in S p {\displaystyle S_{p}} is a copy of C p {\displaystyle C_{p}}. Hence it follows that the number of Sylow p -subgroups is n p = (p − 2) ! {\displaystyle n_{p}=(p-2)!}. The third Sylow theorem implies

Multiplying both sides by (p − 1) gives

that is, the result.
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