Power map permutations and symmetric differences in finite groups

Márton Hablicsek; Guillermo Mantilla-Soler

doi:10.1142/S0219498811005051

paper

Power map permutations and symmetric differences in finite groups

arXiv:1109.2256 · doi:10.1142/S0219498811005051

Abstract

Let $G$ be a finite group. For all $a \in Z$ , such that $(a, ∣ G ∣) = 1$ , the function $ρ_{a} : G \to G$ 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 $ρ_{a}$ . By introducing the group of conjugacy equivariant maps and the symmetric difference method on groups, we exhibit an integer $d_{G}$ such that $sgn (ρ_{a}) = (\frac{d _{G}}{a})$ for all $G$ in a large class of groups, containing all finite nilpotent and odd order groups.

arXiv commentElectronic version of an article to be published as, Journal of Algebra and its Applications, 2011, DOI No: 10.1142/S0219498811005051, \c{opyright} copyright World Scientific Publishing Company, http://www.worldscinet.com/jaa/jaa.shtml