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

• • 上一篇    下一篇

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

PDF (PC)

255

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
[1] KEGEL O H. Sylow-gruppen and subnormalteiler endlicher gruppen[J]. Math Z, 1962, 78(1): 205-221. DOI: 10.1007/BF01195169.
[2] DESKINS W E. On quasinormal subgroups of finite groups[J]. Math Z, 1963, 82(2): 125-132. DOI: 10.1007/BF01111801.
[3] ASAAD M. On the solvability of finite groups[J]. Arch der Math, 1988, 51(4): 289-293. DOI: 10.1007/BF01194016.
[4] SCHMID P. Subgroups permutable with all Sylow subgroups[J]. J Algebra, 1998, 207(1): 285-293. DOI: 10.1006/jabr.1998.7429.
[5] 张勤海,王丽芳. s-半置换子群对群构造的影响[J]. 数学学报, 2005, 48(1): 81-88.
[6] 裴旭莲,钟祥贵. 共轭置换子群与群的幂零性[J]. 广西师范大学学报(自然科学版), 2007, 25(3): 28-31. DOI: 10.16088/j.issn.1001-6600.2007.03.001.
[7] 卢家宽,孟伟. 有限群的π-拟正规子群[J]. 广西师范大学学报(自然科学版), 2011, 29(1): 35-37. DOI: 10.16088/j.issn.1001-6600.2011.01.034.
[8] BALLESTER-BOLINICHES A, WANG Yanming, GUO Xiuyun. C-supplemented subgroups of finite groups[J]. Glasgow Math J, 2000, 42(3): 383-389. DOI: 10.1017/S001708950003007X.
[9] SKIBA A N. On weakly s-permutable subgroups of finite groups[J]. J Algebra, 2007, 315(1): 192-209. DOI: 10.1016/j.jalgebra.2007.04.025.
[10] WANG Yanming. Finite groups with some subgroups of Sylow subgroups c-supplemented[J]. J Algebra, 2000, 224(2): 467-478. DOI: 10.1006/jabr.1999.8079.
[11] ASAAD M, RAMADAN M. Finite groups whose minimal subgroups are c-supplemented[J]. Comm Algebra, 2008, 36(3): 1034-1040. DOI: 10.1080/00927870701776805.
[12] LI Yangming, LI Baojun. On minimal weakly s-supplemented subgroups of finite groups[J]. Journal of Algebra and Its Applications, 2011, 10(5):811-820. DOI: 10.1142/S021949881100494X.
[13] HELIEL A A. A note on c-supplemented subgroups of finite groups[J]. Comm Algebra, 2014, 42(4): 1650-1656. DOI: 10.1080/00927872.2012.747599.
[14] HUPPERT B. Endliche gruppen I[M]. Berlin: Springer-Verlag, 1967. DOI:10.1007/978-3-642-64981-3.
[15] QIAO Shouhong. Weakly s-supplemented subgroups and p-nilpotency of finite groups[J]. Indian Journal of Pure and Applied Mathematics, 2013, 44(6): 795-808. DOI: 10.1007/s13226-013-0043-6.
[16] DOERK K, HAWKES T. Finite soluble groups[M]. Berlin: Walter de Gruyter, 1992.
[17] GURALNICK R M. Subgroups of prime power index in a simple group[J]. J Algebra, 1983, 81(2): 304-311. DOI: 10.1016/0021-8693(83)90190-4.
No related articles found!
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
版权所有 © 广西师范大学学报（自然科学版）编辑部
地址：广西桂林市三里店育才路15号　邮编：541004
电话：0773-5857325　E-mail: gxsdzkb@mailbox.gxnu.edu.cn
本系统由北京玛格泰克科技发展有限公司设计开发