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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.