## Permutation of Z_p

Consider a prime number . Suppose and are permutations of elements of . Prove that can never be a permutation of the elements of .

Consider the product of the elements . Wilson’s theorem says that the product of all the non-zero elements of is equal to . Suppose that the products do form a partition of . 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 by Wilson’s theorem, but in the same time the product should be equal to , being the square of the product of all non-zero elements of . This is a contradiction.

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Categories: IMO, Number theory, Olympiad, Problem Solving
miklos schweitzer

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