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Algebra, 2018], the authors put forward the following conjecture: \\textbf{Conjecture.} \\textit{If $G$ is a group of order $n$ and $\\psi(G)>211\\psi(C_n)/1617 $, where $C_n$ is the cyclic group of order $n$, then $G$ is solvable.} In this paper we prove the validity of this conjecture."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1808.00253","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2018-08-01T10:29:48Z","cross_cats_sorted":[],"title_canon_sha256":"ba199942536ca187d1975092c3e7dff430519aea84012953c2b87e7631215254","abstract_canon_sha256":"f4900caab9de9fe4b410991e9f3e4d13e9448ad25aab29a3fee1cea656a65fc8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:09:06.696641Z","signature_b64":"GshS3m9Lt4l6LD7AZhRAqtvNzAzS/WC3gC4ZlGw5agkO4FP/1d7gAoyqeVaZNR4vGlK1DdE59nsssjO6QoAUBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5d14cb9b888f6b53fe902adbb2d3259a1949b090c56e9ca21f794068429c0acc","last_reissued_at":"2026-05-18T00:09:06.696127Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:09:06.696127Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Criterion for Solvability of a Finite Group by the Sum of Element Orders","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Behrooz Khosravi, Morteza Baniasad Azad","submitted_at":"2018-08-01T10:29:48Z","abstract_excerpt":"Let $G$ be a finite group and $\\psi(G) = \\sum_{g \\in G} o(g)$, where $o(g)$ denotes the order of $g \\in G$. 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