Question 95
Let $p$ be a prime number. For any permutation $\sigma \in S_p$, we define the matrix $A_\sigma = (a_{ij})_{i,j=1,p} \in \mathcal{M}_p(\mathbb{Z})$, whose entries are $a_{ij} = \sigma^{i-1}(j)$, for any $i, j = 1, p$, where $\sigma^0$ is the identity permutation, and $\sigma^k = \underbrace{\sigma \circ \sigma \circ \dots \circ \sigma}_{\text{k times}}$, for any $k \in \mathbb{N}^*$. Prove that the set $D = \{|\det(A_\sigma)| : \sigma \in S_p\}$ has at most $1 + (p-2)!$ elements.