Power map permutations and symmetric differences in finite groups

TL;DR

This paper investigates the signatures of power map permutations in finite groups, linking them to quadratic residues and introducing new algebraic tools to analyze their properties.

Contribution

It introduces the group of conjugacy equivariant maps and the symmetric difference method to relate permutation signatures to quadratic residues in finite groups.

Findings

Established an integer $d_G$ for which the permutation signature equals a quadratic residue symbol.

Extended results to all finite nilpotent and odd order groups.

Provided a new algebraic framework for analyzing power map permutations.

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.