group theoretic proof of Wilson’s theorem

Here we present a group theoretic proof of it.
Clearly, it is enough to show that (p-2)!1(modp) since p-1-1(modp). By Sylow theoremsMathworldPlanetmath, we have that p-Sylow subgroups of Sp, the symmetric groupMathworldPlanetmathPlanetmath on p elements, have order p, and the number np of Sylow subgroups is congruent to 1 modulo p. Let P be a Sylow subgroup of Sp. Note that P is generated by a p-cycle. There are (p-1)! cycles of length p in Sp. Each p-Sylow subgroup contains p-1 cycles of length p, hence there are (p-1)!p-1=(p-2)! different p-Sylow subgrups in Sp, i.e. nP=(p-2)!. From Sylow’s Second Theorem, it follows that (p-2)!1(modp),so (p-1)!-1(modp).

Title group theoretic proof of Wilson’s theorem
Canonical name GroupTheoreticProofOfWilsonsTheorem
Date of creation 2013-03-22 13:35:27
Last modified on 2013-03-22 13:35:27
Owner ottocolori (1519)
Last modified by ottocolori (1519)
Numerical id 10
Author ottocolori (1519)
Entry type Proof
Classification msc 11-00