Power map permutations and symmetric differences in finite groups

Abstract

Let $G$ be a finite group. For all $a \in \Z$, such that $(a,|G|)=1$, the function $\rho_a: G \to G$ sending $g$ to $g^a$ defines a permutation of the elements of $G$. Motivated by a recent generalization of Zolotarev's proof of classic quadratic reciprocity, due to Duke and Hopkins, we study the signature of the permutation $\rho_a$. By introducing the group of conjugacy equivariant maps and the symmetric difference method on groups, we exhibit an integer $d_{G}$ such that $\text{sgn}(\rho_a)=(\frac{d_G}{a})$ for all $G$ in a large class of groups, containing all finite nilpotent and odd order groups.