每对非共轭极大子群之积构成的有限群是可解群
Finite groups that are the product of every pair of non-conjugate maximal subgroups are soluble
AI总结：
该研究证明每对非共轭极大子群之积构成的有限群必可解，否定解决了Kourovka笔记的问题10.34，通过可除性准则及GAP验证消除一般情形的基S^k障碍。
AI中文摘要：
我们证明：若有限群等于其任意两个非共轭极大子群的乘积，则该群是可解群，从而否定性解决了《Kourovka 笔记》中的问题10.34（V. S. Monakhov，1986）。几乎单群情形已由Tikhonenko与Tyutyanov（2010）解决；一般情形的障碍是基S^k（k≥2），计数界无法控制该情形。我们通过一个可除性准则消除该障碍：对单群S的自同构稳定子群共轭类的单一对，满足赋值不等式，可一次性排除所有k≥2及所有容许嵌入的基S^k。对有限单群的每个无限族，我们以Zsigmondy素数作为算术障碍，统一构造此类对子；散在单群及所有剩余小情形，通过GAP生成的可独立核验的验证集解决。
英文摘要：
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.
