{"paper":{"title":"On the Normalizer-Solubilizer Conjecture_V3","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Hamid Mousavi","submitted_at":"2025-01-20T13:38:00Z","abstract_excerpt":"Let $G$ be a finite group and $x$ be an element of $G$. Define $\\textrm{Sol}_G(x)$ as the set of all $y \\in G$ such that $\\langle {x,y}\\rangle$ is soluble. We provide an equivalent condition for the normalizer-solubilizer conjecture, namely $|\\mathcal{N}_G(\\langle x\\rangle)| \\mid |\\textrm{Sol}_G(x)|$, where $\\mathcal{N}_G(\\langle x\\rangle)$ is the normalizer of $\\langle x\\rangle$. Furthermore, we demonstrate that the conjecture holds in the special case where $\\mathcal{N}_G(\\langle x\\rangle)$ is a Frobenius group with kernel $\\mathcal{C}_G(x)$, the centralizer of $x$, and $|\\mathcal{N}_G(\\lang"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2501.11486","kind":"arxiv","version":3},"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/2501.11486/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"}