{"paper":{"title":"Power map permutations and symmetric differences in finite groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Guillermo Mantilla-Soler, M\\'arton Hablicsek","submitted_at":"2011-09-10T19:45:43Z","abstract_excerpt":"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 orde"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.2256","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}