On an extension of Zolotarev's lemma and some permutations
Liyuan Wang, Hai-Liang Wu · arXiv (Cornell University) · 2018
Let $p$ be an odd prime, for each integer $a$ with $p mid a$, the famous Zolotarev's Lemma says that the Legendre symbol $(\frac{a}{p})$ is the sign of the permutation of $\Z/p\Z$ induced by multiplication by $a$. Then Frobenius extended Zolotarev's result to all positive odd integers. Recently, Sun \cite{S} study the permutation problems involving quadratic residues. Motivated by the above work, in this paper, we first extend the result of Frobenius to all positive integers. In addition, we discuss some permutation problems involving quadratic residues modulo an odd prime $p$. In particular, we confirm some conjectures posed by Sun. Finally, we study the permutation problems induced by primitive root of a power of an odd prime $p$.