广西师范大学学报（自然科学版） ›› 2017, Vol. 35 ›› Issue (1): 49-52.doi: 10.16088/j.issn.1001-6600.2017.01.008

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子群弱s-可补性对有限群可解性的影响

卢家宽,邱燕燕   

  1. 广西师范大学数学与统计学院,广西桂林541004
  • 出版日期:2017-01-20 发布日期:2018-07-17
  • 通讯作者: 卢家宽(1980—),男(壮族),广西贵港人,广西师范大学副教授。E-mail:jklu@gxnu.edu.cn
  • 基金资助:
    国家自然科学基金(11461007);广西高校数学与统计模型重点实验室开放基金(2016GXKLMS002)

The Influence of Weakly s-supplemented Subgroups on Solvability of Finite Groups

LU Jiakuan, QIU Yanyan   

  1. College of Mathematics and Statistics, Guangxi Normal University, Guilin Guangxi 541004, China
  • Online:2017-01-20 Published:2018-07-17

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摘要： 称有限群G的子群H为弱s-可补子群,如果存在G的子群K,使得G=HK,H∩K≤HsG,其中HsG是由H的在G中s-置换的子群生成的子群。本文证明了如下结果:①如果有限群G的奇素数阶子群在G中弱s-可补,那么G是可解群;②有限群G是可解群当且仅当G的所有奇数阶Sylow子群在G中弱s-可补。这2个结果推广和改进了已有的结果。

关键词: 弱s-可补子群; 可解群; 超可解群

Abstract: A subgroup H of a finite group G is called weakly s-supplemented subgroup of G if there is a subgroup K of G such that G=HK and H∩K≤HsG, where HsG is the subgroup of H generated by all those subgroups of H which are s-permutable in G. In this paper, the following results are proved: (1) a finite group G is solvable if every subgroup of odd prime order of G is weakly s-supplemented in G; (2) a finite group G is solvable if and only if every Sylow subgroup of odd order of G is weakly s-supplemented in G. Some classical and recent results are extended.

Key words: weakly s-supplemented subgroup, solvable group, supersolvable group

中图分类号: 

  • O152.1
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