{"paper":{"title":"An exact upper bound for sums of element orders in non-cyclic finite groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Marcel Herzog, Mercede Maj, Patrizia Longobardi","submitted_at":"2016-10-12T11:07:28Z","abstract_excerpt":"Denote the sum of element orders in a finite group $G$ by $\\psi(G)$ and let $C_n$ denote the cyclic group of order $n$. Suppose that $G$ is a non-cyclic finite group of order $n$ and $q$ is the least prime divisor of $n$. We proved that $\\psi(G)\\leq\\frac 7{11}\\psi(C_n)$ and $\\psi(G)<\\frac 1{q-1}\\psi(C_n)$. The first result is best possible, since for each $n=4k$, $k$ odd, there exists a group $G$ of order $n$ satisfying $\\psi(G)=\\frac 7{11}\\psi(C_n)$ and the second result implies that if $G$ is of odd order, then $\\psi(G)<\\frac 12\\psi(C_n)$. Our results improve the inequality $\\psi(G)<\\psi(C_n"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.03669","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"}