Editing Wilson's theorem (section)

====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]] <math> S_p </math> has exactly <math>(p-1)!</math> elements of order ''p'', namely the ''p''-cycles <math> C_p </math>. On the other hand, each Sylow ''p''-subgroup in <math> S_p </math> is a copy of <math> C_p </math>. Hence it follows that the number of Sylow ''p''-subgroups is <math> n_p=(p-2)! </math>. The third Sylow theorem implies
:<math>(p-2)! \equiv 1 \pmod p.</math>
Multiplying both sides by {{nowrap|1=(''p'' − 1)}} gives
:<math>(p-1)! \equiv p-1 \equiv -1 \pmod p,</math>
that is, the result.