极大不变子群与有限群的可解性  

Maximal Invariant Subgroups and Solvability of Finite Groups

作　　者：刘玉萍 李明晖 庞琳娜 卢家宽 LIU Yu-ping;LI Ming-hui;PANG Lin-na;LU Jia-kuan(School of Mathematics and Statistics,Guangxi Normal University,Guangxi Guilin 541006)

机构地区：[1]广西师范大学数学与统计学院,广西桂林541006

出　　处：《河北北方学院学报（自然科学版）》2023年第5期1-3,6,共4页Journal of Hebei North University:Natural Science Edition

基　　金：国家自然科学基金项目(11861015);广西研究生教育创新计划(XYCSZ2021011)。

摘　　要：目的讨论在互素作用下群G的极大A-不变子群及其A-不变Sylow子群的次正规性与S-半正规性对有限群结构的影响。方法使用极小阶反例法和归纳法。结果得到了有限群G是可解群或Sylow塔群的若干充分条件。结论在互素作用下,若有限群G存在极大A-不变子群的A-不变Sylow子群是S-半正规子群,则群G是可解群;若有限群G的2-极大A-不变子群是次正规子群,则G是Sylow塔群。Objective To discuss the influence of subnormality and S-seminormality of maximal A-invariant subgroups or its A-invariant Sylow subgroups on the structure of finite groups under the coprime action.Methods Minimal order counterexample or induction.Results Some sufficient conditions for a finite group G to is a solvable group or a Sylow tower group are obtained.Conclusion Let a finite group A act coprimely on a finite group G by automorphisms.Suppose that there exists a maximal A-invariant subgroup M of G such that the A-invariant Sylow subgroups of M are S-seminormal subgroups,then G is solvable.If the 2-maximal A-invariant subgroups of G are subnormal subgroups,then G is a Sylow tower group.

关 键 词：互素作用 次正规子群 可解群 

分 类 号：O152.1[理学—数学]