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%3 N_20200518114452a_ColinWright    By ColinWright 2020/05/18 @ 11:44:52a -------------------------------- Firstly, throughout we will assume p is prime, greater than 2, and we will work modulo p.        (select only this node)     N_20200518114452b_ColinWright    By ColinWright 2020/05/18 @ 11:44:52b -------------------------------- For every x non-zero, there is a y such that xy=1.        (select only this node)     N_20200518114452a_ColinWright->N_20200518114452b_ColinWright N_20200207132655d_CDW    By CDW 2020/02/07 @ 13:26:55d -------------------------------- Thus p cannot be 4k+3.        (select only this node)     N_20200207111612a_CDW    By CDW 2020/02/07 @ 11:16:12a -------------------------------- Group Theory for Fun and Profit        (select only this node)     N_20200207111612d_CDW    By CDW 2020/02/07 @ 11:16:12d -------------------------------- Groups        (select only this node)     N_20200207111612a_CDW->N_20200207111612d_CDW N_20200207133642a_CDW    By CDW 2020/02/07 @ 13:36:42a -------------------------------- There's one group of order 1.        (select only this node)     N_20200207133642b_CDW    By CDW 2020/02/07 @ 13:36:42b -------------------------------- There's one group of order 2.        (select only this node)     N_20200207133642a_CDW->N_20200207133642b_CDW N_20200207111612b_CDW    By CDW 2020/02/07 @ 11:16:12b -------------------------------- Lagrange's Theorem        (select only this node)     N_20200207122821a_CDW    By CDW 2020/02/07 @ 12:28:21a -------------------------------- If p=4k+3 then "-1" does *not* have a square root.        (select only this node)     N_20200207111612b_CDW->N_20200207122821a_CDW N_20200207133642c_CDW    By CDW 2020/02/07 @ 13:36:42c -------------------------------- There's one group of order 3.        (select only this node)     N_20200207133642b_CDW->N_20200207133642c_CDW N_20200207111612c_CDW    By CDW 2020/02/07 @ 11:16:12c -------------------------------- Modulo Arithmetic        (select only this node)     N_20200207111612i_CDW    By CDW 2020/02/07 @ 11:16:12i -------------------------------- Square root of -1        (select only this node)     N_20200207111612c_CDW->N_20200207111612i_CDW N_20200207111612k_CDW    By CDW 2020/02/07 @ 11:16:12k -------------------------------- Inverse of an element        (select only this node)     N_20200207111612c_CDW->N_20200207111612k_CDW N_20200207133642d_CDW    By CDW 2020/02/07 @ 13:36:42d -------------------------------- There are exactly two groups that have exactly 4 elements.        (select only this node)     N_20200207133642c_CDW->N_20200207133642d_CDW N_20200207120632a_CDW    By CDW 2020/02/07 @ 12:06:32a -------------------------------- Cayley Tables        (select only this node)     N_20200207111612d_CDW->N_20200207120632a_CDW N_20200207113841a_CDW    By CDW 2020/02/07 @ 11:38:41a -------------------------------- Commutative        (select only this node)     N_20200207111612d_CDW->N_20200207113841a_CDW N_20200207113841b_CDW    By CDW 2020/02/07 @ 11:38:41b -------------------------------- Non-Commutative        (select only this node)     N_20200207111612d_CDW->N_20200207113841b_CDW N_20200207120632b_CDW    By CDW 2020/02/07 @ 12:06:32b -------------------------------- (Z_4,+)        (select only this node)     N_20200207133642d_CDW->N_20200207120632b_CDW N_20200207120642a_CDW    By CDW 2020/02/07 @ 12:06:42a -------------------------------- K_4        (select only this node)     N_20200207133642d_CDW->N_20200207120642a_CDW N_20200207111612e_CDW    By CDW 2020/02/07 @ 11:16:12e -------------------------------- Permutation Group        (select only this node)     N_20200207111612h_CDW    By CDW 2020/02/07 @ 11:16:12h -------------------------------- Permutations on four elements : S_4        (select only this node)     N_20200207111612e_CDW->N_20200207111612h_CDW N_20200518114452c_ColinWright    By ColinWright 2020/05/18 @ 11:44:52c -------------------------------- We never have x^2=1, so x and y are different.        (select only this node)     N_20200518114452b_ColinWright->N_20200518114452c_ColinWright N_20200207111612f_CDW    By CDW 2020/02/07 @ 11:16:12f -------------------------------- Symmetry Group        (select only this node)     N_20200207111612g_CDW    By CDW 2020/02/07 @ 11:16:12g -------------------------------- Cube        (select only this node)     N_20200207111612f_CDW->N_20200207111612g_CDW N_20200207114505a_CDW    By CDW 2020/02/07 @ 11:45:05a -------------------------------- Tetrahedron        (select only this node)     N_20200207111612f_CDW->N_20200207114505a_CDW N_20200518114452d_ColinWright    By ColinWright 2020/05/18 @ 11:44:52d -------------------------------- We can pair off all the number from 2 to (p-2), each with its inverse.        (select only this node)     N_20200518114452c_ColinWright->N_20200518114452d_ColinWright N_20200518114605a_ColinWright    By ColinWright 2020/05/18 @ 11:46:05a -------------------------------- Suppose x^2=1. Then x^2-1=0, and hence (x-1)(x+1)=0. From that we can conclude that x=1 or x=-1.        (select only this node)     N_20200518114452c_ColinWright->N_20200518114605a_ColinWright N_20200207114900a_CDW    By CDW 2020/02/07 @ 11:49:00a -------------------------------- Same number of elements. Hmm ...        (select only this node)     N_20200207111612g_CDW->N_20200207114900a_CDW N_20200207115430a_CDW    By CDW 2020/02/07 @ 11:54:30a -------------------------------- Finitely many elements - finite groups        (select only this node)     N_20200207111612g_CDW->N_20200207115430a_CDW N_20200518114452e_ColinWright    By ColinWright 2020/05/18 @ 11:44:52e -------------------------------- So the numbers 2 to (p-2) multiply out to give 1.        (select only this node)     N_20200518114452d_ColinWright->N_20200518114452e_ColinWright N_20200207111612h_CDW->N_20200207115430a_CDW N_20200207114822a_CDW    By CDW 2020/02/07 @ 11:48:22a -------------------------------- Subgroup        (select only this node)     N_20200207111612h_CDW->N_20200207114822a_CDW N_20200207111612h_CDW->N_20200207114900a_CDW N_20200518114452f_ColinWright    By ColinWright 2020/05/18 @ 11:44:52f -------------------------------- So (p-1)!=-1 (mod p)        (select only this node)     N_20200518114452e_ColinWright->N_20200518114452f_ColinWright N_20200207111612i_CDW->N_20200207122821a_CDW N_20200207111612j_CDW    By CDW 2020/02/07 @ 11:16:12j -------------------------------- Wilson's Theorem        (select only this node)     N_20200207111612i_CDW->N_20200207111612j_CDW N_20200207122638a_CDW    By CDW 2020/02/07 @ 12:26:38a -------------------------------- If p=4k+1, then "-1" has a square root.        (select only this node)     N_20200207111612i_CDW->N_20200207122638a_CDW N_20200207111612j_CDW->N_20200518114452a_ColinWright N_20200207111612j_CDW->N_20200207122638a_CDW N_20200207111612l_CDW    By CDW 2020/02/07 @ 11:16:12l -------------------------------- Z_p        (select only this node)     N_20200207111612k_CDW->N_20200207111612l_CDW N_20200207111612k_CDW->N_20200207111612j_CDW N_20200207111612l_CDW->N_20200207115430a_CDW N_20200207114015c_CDW    By CDW 2020/02/07 @ 11:40:15c -------------------------------- Non-zero rationals with multiplication        (select only this node)     N_20200207113841a_CDW->N_20200207114015c_CDW N_20200207114015a_CDW    By CDW 2020/02/07 @ 11:40:15a -------------------------------- Integers with addition        (select only this node)     N_20200207113841a_CDW->N_20200207114015a_CDW N_20200207115739a_CDW    By CDW 2020/02/07 @ 11:57:39a -------------------------------- Integers with multiplication are not a group ... there is no inverse.        (select only this node)     N_20200207113841a_CDW->N_20200207115739a_CDW N_20200207114015b_CDW    By CDW 2020/02/07 @ 11:40:15b -------------------------------- Positive rationals with multiplication        (select only this node)     N_20200207113841a_CDW->N_20200207114015b_CDW N_20200207113841b_CDW->N_20200207111612f_CDW N_20200207113841b_CDW->N_20200207111612e_CDW N_20200207115337a_CDW    By CDW 2020/02/07 @ 11:53:37a -------------------------------- Infinitely many elements - an infinite group        (select only this node)     N_20200207114015a_CDW->N_20200207115337a_CDW N_20200207114152a_CDW    By CDW 2020/02/07 @ 11:41:52a -------------------------------- Subgroup        (select only this node)     N_20200207114015b_CDW->N_20200207114152a_CDW N_20200207114015b_CDW->N_20200207115337a_CDW N_20200207114015c_CDW->N_20200207114152a_CDW N_20200207114015c_CDW->N_20200207115337a_CDW N_20200207114505a_CDW->N_20200207115430a_CDW N_20200207114505a_CDW->N_20200207114822a_CDW N_20200207114822a_CDW->N_20200207111612b_CDW N_20200207115003a_CDW    By CDW 2020/02/07 @ 11:50:03a -------------------------------- Are they, in some sense, the "Same Group"?        (select only this node)     N_20200207114900a_CDW->N_20200207115003a_CDW N_20200207115430a_CDW->N_20200207111612b_CDW N_20200207115739a_CDW->N_20200207111612c_CDW N_20200207120632a_CDW->N_20200207133642a_CDW N_20200207120632a_CDW->N_20200207133642b_CDW N_20200207120632a_CDW->N_20200207133642d_CDW N_20200207120632a_CDW->N_20200207133642c_CDW N_20200207120632b_CDW->N_20200207115430a_CDW N_20200207122144a_CDW    By CDW 2020/02/07 @ 12:21:44a -------------------------------- Not the same!        (select only this node)     N_20200207120632b_CDW->N_20200207122144a_CDW N_20200207120642a_CDW->N_20200207122144a_CDW N_20200207120642a_CDW->N_20200207115430a_CDW N_20200207131428a_CDW    By CDW 2020/02/07 @ 13:14:28a -------------------------------- By Wilson's Theorem, (4k)! = -1        (select only this node)     N_20200207122638a_CDW->N_20200207131428a_CDW N_20200207132655a_CDW    By CDW 2020/02/07 @ 13:26:55a -------------------------------- Suppose x^2 = -1        (select only this node)     N_20200207122821a_CDW->N_20200207132655a_CDW N_20200207131428b_CDW    By CDW 2020/02/07 @ 13:14:28b -------------------------------- Observe that d = -(4k-d)        (select only this node)     N_20200207131428a_CDW->N_20200207131428b_CDW N_20200207131428c_CDW    By CDW 2020/02/07 @ 13:14:28c -------------------------------- So (2k+1)(2k+2)...(4k) = -1^(2k).(2k)!        (select only this node)     N_20200207131428b_CDW->N_20200207131428c_CDW N_20200207131428d_CDW    By CDW 2020/02/07 @ 13:14:28d -------------------------------- So (p-1)! = ((2k)!)^2        (select only this node)     N_20200207131428c_CDW->N_20200207131428d_CDW N_20200207131428e_CDW    By CDW 2020/02/07 @ 13:14:28e -------------------------------- Thus {(p-1)/2}^2 = -1, as required.        (select only this node)     N_20200207131428d_CDW->N_20200207131428e_CDW N_20200207132655b_CDW    By CDW 2020/02/07 @ 13:26:55b -------------------------------- Then { 1, x, x^2, x^3 } is a subgroup.        (select only this node)     N_20200207132655a_CDW->N_20200207132655b_CDW N_20200207132655c_CDW    By CDW 2020/02/07 @ 13:26:55c -------------------------------- So 4 divides the order of the group.        (select only this node)     N_20200207132655b_CDW->N_20200207132655c_CDW N_20200207132655c_CDW->N_20200207132655d_CDW