{"paper":{"title":"A compensation theorem for the Sylow-integral invariant and counterexamples to an \\texorpdfstring{$A_5$}{A5}-characterization conjecture","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Yaoran Yang, Yutong Zhang","submitted_at":"2026-05-20T10:06:19Z","abstract_excerpt":"Let \\(\\nu_p(G)\\) be the number of Sylow \\(p\\)-subgroups of a finite group \\(G\\), let \\(\\sigma_p(G)\\) be their common order, and set \\[\n  \\gamma(G)=\\int_0^1\\sum_{p\\in\\pi(G)}\\nu_p(G)x^{\\sigma_p(G)}\\,dx\n  =\\sum_{p\\in\\pi(G)}\\frac{\\nu_p(G)}{\\sigma_p(G)+1}. \\] A recent conjectural extension of the simple-group theorem for this invariant asserted that a nonsolvable finite group has \\(\\gamma(G)=9/2\\) precisely when \\(G\\cong A_5\\). We disprove this assertion by a direct and verifiable construction. More generally, we prove an exact direct-product compensation formula for \\(A_5\\) with an arbitrary nilpo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.20976","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.20976/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}