  Let $\FF$ be a finite field of characteristics different from two. We show
that no bijective map transforms permanent into determinant when the
cardinality of $\FF$ is sufficiently large. We also give an example of
non-bijective map when $\FF$ is arbitrary and an example of a bijective map
when $\FF$ is infinite which do transform permanent into determinant. The
developed technique allows us to estimate the probability of the permanent and
the determinant of matrices over finite fields to have a given value. Our
results are also true over finite rings without zero divisors.
