Let $R_n$ be the number of reduced Latin squares of order $n$. Then $R_n \equiv 1 \pmod n$ if $n$ is prime and $R_n \equiv 0 \pmod n$ if $n$ is composite. See our paper Divisors of the number of Latin rectangles.

In this case, when $n=p$ is a prime, this is arrived at via a group action (by a group of cardinality $p$) that has $(p-2)!$ fixed points. Wilson's Theorem implies $(p-2)! \equiv 1 \pmod p$, giving the result.