作者:田云凤， 史江涛， 刘文静
Authors:\n\tTIANYunfeng,\tSHIJiangtao,\tLIUWenjing\n
摘要: 为了进一步研究每个非幂零极大子群的指数皆为素数的有限群的可解性,使用反证法和极小阶反例的方法,并结合应用 Wielandt 给出的一个关于具有幂零 Hall - 子群( 不是 Sylow - 子群) 的有限群 G 的结构刻画的定理,得到了一个较为初等的关于每个非幂零极大子群的指数皆为素数的有限群 G 的可解性的证明。 该证明没有应用 Glauberman-Thompson p - 幂零准则和 Rose 的关于具有幂零极大子群的非交换单群的分类和关于具有幂零极大子群且中心等于 1 的非可解群的刻画,这改进了之前在相关的研究文献中关于这个结论的证明。

Abstract: In order to give a further study of the solvability of a finite group in which every non-nilpotent maximal subgroup has prime index, the methods of the proof by contradiction and the counterexample of the smallest order and a theorem of Wielandt on the characterization of the structure of a finite group G with a nilpotent Hall-subgroup which is not a Sylow subgroup are applied to obtain a more elementary proof of the solvability of a finite group in which every non-nilpotent maximal subgroup has prime index. The proof does not apply the Glauberman-Thompson p-nilpotent criterion and Rose′s two results on a classification of non-abelian simple groups with nilpotent maximal subgroup and a characterization of non-solvable group with nilpotent maximal subgroup and trivial center respectively, which improves the proof of the result in the relevant research references.

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