 Solution

For prime $n$, pair each nonzero residue with its inverse; only $\pm 1$ are self-inverse, yielding Wilson theorem. Conversely, if $(n-1)!\equiv -1(\mathrm{mod}n)$, every $1\le a<n$ is a unit (otherwise a common divisor would divide the left side and $n$ but not $-1$), so $n$ is prime.