{"paper":{"title":"Termination of the Lattice-Automorphism Tower for Direct Products of Symmetric Groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Sonukumar, Vinay Madhusudanan","submitted_at":"2026-05-20T11:00:00Z","abstract_excerpt":"Let $G$ be a finite group. Let $\\mathcal{N}(G)$ be the lattice of normal subgroups ordered by inclusion, regarded as an abstract lattice. Define $\\operatorname{LatAut}(G) := \\operatorname{Aut}(\\mathcal{N}(G))$. The \\emph{LatAut tower} is the sequence defined by $G_0 = G$, $G_{n+1} = \\operatorname{LatAut}(G_n)$.\n  Let $G$ be a \\emph{tower group} if $G \\cong \\prod_{k \\geq 3} S_k^{a_k}$ with finitely many $a_k \\neq 0$. We establish the following for tower groups.\n  \\emph{Product Formula.} $\\operatorname{LatAut}\\!\\bigl(\\prod_{k \\geq 3} S_k^{a_k}\\bigr) \\cong S_{a_4} \\times S_B$, where $B = \\sum_{k "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.21025","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/2605.21025/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"}