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Permutation of Z_p


Consider a prime number p > 2. Suppose a_1,a_2,...,a_p and b_1,b_2,...,b_p are permutations of elements of \Bbb{Z}_p. Prove that a_1b_1,...,a_pb_p can never be a permutation of the elements of \Bbb{Z}_p.

Consider the product of the elements a_ib_i. Wilson’s theorem says that the product of all the non-zero elements of \Bbb{Z}_p is equal to -1. Suppose that the products do form a partition of \Bbb{Z}_p. Then obviously zero must be paired with zero, or else we would have two zeros. The product of the rest of the elements should be equal to -1 by Wilson’s theorem, but in the same time the product should be equal to (-1)^2, being the square of the product of all non-zero elements of \Bbb{Z}_p. This is a contradiction.

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