group theoretic proof of Wilson’s theorem
Here we present a group theoretic proof of it.
Clearly, it is enough to show that since . By Sylow theorems, we have that -Sylow subgroups of , the symmetric group on elements, have order , and the number of Sylow subgroups is congruent to 1 modulo . Let be a Sylow subgroup of . Note that is generated by a -cycle. There are cycles of length in . Each -Sylow subgroup contains cycles of length , hence there are different -Sylow subgrups in , i.e. . From Sylow’s Second Theorem, it follows that ,so .
|Title||group theoretic proof of Wilson’s theorem|
|Date of creation||2013-03-22 13:35:27|
|Last modified on||2013-03-22 13:35:27|
|Last modified by||ottocolori (1519)|