非共轭极大子群的乘积
Products of nonconjugate maximal subgroups
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中文总结 AI 辅助
该研究证明了满足任意一对非共轭极大子群乘积为群本身的有限群必可解，否定解决了库罗夫卡问题集的10.34号问题，并应用该结果解答了Guo提出的一个公开问题。
中文摘要 AI 辅助
我们证明：若有限群$G$的每一对非共轭极大子群$M,N<G$都满足$MN=G$，则$G$是可解群。等价地，每个有限非可解群都存在两个非共轭极大子群，它们的集合乘积是真子群。该结果否定回答了《Kourovka Notebook（库罗夫卡问题集）》的问题10.34。作为应用，我们还解答了Guo提出的一个公开问题。
英文摘要
We prove that a finite group $G$ is solvable whenever $MN=G$ for every pair of nonconjugate maximal subgroups $M,N<G$. Equivalently, every finite nonsolvable group has two nonconjugate maximal subgroups whose setwise product is proper. This gives a negative answer to Problem 10.34 of the Kourovka Notebook. As an application, we also answer an open question raised by Guo.
