{"paper":{"title":"Sums of element orders in groups of odd order","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":"2019-05-29T09:42:30Z","abstract_excerpt":"Denote by $G$ a finite group and by $\\psi(G)$ the sum of element orders in $G$. If $t$ is a positive integer, denote by $C_t$ the cyclic group of order $t$ and write $\\psi(t)=\\psi(C_t)$. In this paper we proved the following Theorem A: Let $G$ be a non-cyclic group of odd order $n=qm$, where $q$ is the smallest prime divisor of $n$ and $(m,q)=1$. Then the following statements hold. (1) If $q=3$, then $\\frac {\\psi(G)}{\\psi(|G|)}\\leq \\frac {85}{301}$, and equality holds if and only if $n=3\\cdot 7\\cdot m_1$ with $(m_1,42)=1$ and $G=(C_7\\rtimes C_3)\\times C_{m_1}$, with $C_7\\rtimes C_3$ non-abelia"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.12291","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"}