A characterization of Sophie Germain primes

Abstract: Let $n\ge 5$ be an odd integer. It is shown that ${1^{\sigma(1)},\ldots,n^{\sigma(n)}}$ is a complete residue system modulo $n$ for some permutation $\sigma$ of ${1,\ldots,n}$ if and only if $\frac{1}{2}(n-1)$ is a Sophie Germain prime. Partial results are obtained also for the case $n$ even.