Finite groups that are the product of every pair of non-conjugate maximal subgroups are soluble

[2608.02970] Finite groups that are the product of every pair of non-conjugate maximal subgroups are soluble Skip to main content Search Submit Donate Log in Mathematics > Group Theory arXiv:2608.02970 (math) [Submitted on 4 Aug 2026 ( v1 ), last revised 27 Aug 2026 (this version, v3)] Title: Finite groups that are the product of every pair of non-conjugate maximal subgroups are soluble Authors: Richie Sater View PDF HTML (experimental) Abstract: We prove that every finite group that is the product of every pair of its non-conjugate maximal subgroups is soluble, answering Problem 10.34 of the Kourovka Notebook. The almost-simple case was proved by Tikhonenko and Tyutyanov. We treat the remaining minimal-counterexample branch, where the unique minimal normal subgroup has the form S^k, with S nonabelian simple and k >= 2. Two automorphism-stable coordinate subgroup classes satisfying a maximal-supplement criterion produce non-conjugate maximal supplements. If the factorization hypothesis held, their product would impose a p-adic divisibility requirement growing linearly with k, while the quotient contributes only coordinate outer automorphisms and a factor dividing k!. A fixed valuation gap therefore excludes every k >= 2 at once. Suitable subgroup classes are constructed uniformly across the infinite families of finite simple groups using parabolic, torus, and primitive-prime-divisor arguments; stable flag parabolics handle graph fusion, while GAP certificates cover designated finite and sporadic cases. The resulting all-k obstruction provides a reusable mechanism for eliminating direct-power socles in finite-group factorization problems. Comments: 26 pages, 4 tables, no figures. Corrected manuscript; abstract, author information, end matter, and exposition revised; results unchanged. Supporting materials: this https URL (v1.1.2); archive: this https URL Subjects: Group Theory (math.GR) MSC classes: 20D40 (Primary) 20E28, 20D05 (Secondary) Cite as: arXiv:2608.02970 [math.GR] (or arXiv:2608.02970v3 [math.GR] for this version) https://doi.org/10.48550/arXiv.2608.02970 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Richie Sater [ view email ] [v1] Tue, 4 Aug 2026 00:06:17 UTC (30 KB) [v2] Thu, 6 Aug 2026 01:01:45 UTC (30 KB) [v3] Thu, 27 Aug 2026 23:10:53 UTC (35 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.GR < prev | next > new | recent | 2026-08 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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