模糊水平子群与正规模糊水平子群

P. Sivaramakrishna Das~[1]和吴望名~[2]分别引入了Fuzzy水平子群和正规Fuzzy子群等概念及一些基本性质。本文在[1]和[2]的基础上引入正规水平子群,并讨论水平予群和正现水平子群的个数对群所产生的影响。     

有限群的Fuzzy次正规子群与Fuzzy极大子群

本文研究了有限群的Ｆ次正规子群，得出了一个Ｆ子群是Ｆ次正规子群的充要条件，讨论了Ｆ次正规子群的一些重要性质。另外，本文还引入了有限群的Ｆ极大子群的概念，给出了Ｆ子群是Ｆ极大群的充要条件。最后，给出了三个定理，讨论了有限群Ｇ可解、超可解、幂零与Ｇ的Ｆ次正规子群、Ｆ极大子群之间的联系。     

一类可解的（q）－群

有限群Ｇ叫（ｑ）－群，如果Ｇ中每个次正规子群均为拟正规子群，群Ｇ叫Ｅｑ－群，若Ｇ中每个子群在Ｇ中拟正规或自正规，有限群Ｇ叫内Ｅｑ－群，如果Ｇ本身不是Ｅｑ－群，但Ｇ的每个真子群是Ｅｑ－群，本文确定了Ｅｑ－群的结构与内Ｅｑ－群的分类．     

关于TL子群的若干性质

引入拟正规的TL子群的概念，讨论一些重要性质。此外，在M．A．A．Mishref文献[7]和J-G Kim文献[8]的基础上，继续研究TL-p子群。     

只具有半正规和反正规子群的有限群

群Ｇ的子群Ｈ称为是Ｇ的半正规子群，若对任意Ｇ的子群Ｋ，只要Ｋ的阶数与Ｈ互质，则Ｈ与Ｋ可置换；称Ｈ在Ｇ中是反正规的，若对任意ｇ∈Ｇ，ｇ，∈＜Ｈ．Ｈｇ＞．本文刻画了每个子群是半正规的，或反正规的有限群，推广了［１］及［２］中的结果．     

TH型区间值模糊正规子群

本文在区间值模糊集空间上，引入了幂等区间范数ＴＨ，在此基础上，定义了ＴＨ型区间值模糊正规子群，并研究了它的一些性质和结构特征，从而拓广了区间值模糊集的理论。     

模糊群的若干性质

本文证明了下述结果：设μ是群Ｇ的模糊子群，那么１，若ν（ｘｙ＾－１）≥ｍａｘ（μ（ｘ），μ（ｙ），μ（ｘ）＝μ（ｙ）；２。若ｘμ＝ｙμ，则μ（ｘ）＝μ（ｙ）；３．μ是模糊正规子群当且仅当μｖ＝ｖμ。４本文指出［３］的命题２与定理６的证明都是错误的，但定理６的结论是正确的，本文给出一新的证明。     

群G的反模糊子群

给出一个群G的反模糊子群和正规反模糊子群的概念，这些定义不同于Rosenfeld和吴望名等的定义，本文还讨论了正规反模糊子群的一些性质及模糊商群等。     

一类可补子群

本文在有限群的半正规子群的研究基础上对可补子群进行研究．用半正规代替正则，推广了已有的一些结果．     

Finite groups with hall subnormally embedded Schmidt subgroups

Abstract

A subgroup H of a finite group G is said to be Hall subnormally embedded in G if there is a subnormal subgroup N of G such that H is a Hall subgroup of N. A Schmidt group is a finite non-nilpotent group whose all proper subgroups are nilpotent. We prove the nilpotency of the second derived subgroup of a finite group in which each Schmidt subgroup is Hall subnormally embedded.     

Finite groups with Hall subnormally embedded Schmidt subgroups

Abstract

A subgroup H of a finite group G is said to be Hall subnormally embedded in G if there is a subnormal subgroup N of G such that H is a Hall subgroup of N. A Schmidt group is a finite non-nilpotent group whose all proper subgroups are nilpotent. We prove the nilpotency of the second derived subgroup of a finite group in which each Schmidt subgroup is Hall subnormally embedded.     

某些子群为S-半正规的有限群

A subgroup H is called S-seminormal in a finite group G if H permutes with all Sylow p-subgroups of G with (p, |H| =1. The main object of this paper is to generalize some known results about finite supersolvable groups to a saturated formation containing the class of finite supersolvable groups.     

Finite Groups with only S-seminormal or Selfnormal Subgroups

Ｆｉｎｉｔｅ　Ｇｒｏｕｐｓ　ｗｉｔｈ　ｏｎｌｙ　Ｓ－ｓｅｍｉｎｏｒｍａｌ　ｏｒ　Ｓｅｌｆｎｏｒｍａｌ　ＳｕｂｇｒｏｕｐｓＦｉｎｉｔｅＧｒｏｕｐｓｗｉｔｈｏｎｌｙＳ－ｓｅｍｉｎｏｒｍａｌｏｒＳｅｌｆｎｏｒｍａｌＳｕｂｇｒｏｕｐｓ￥ＺｈａｎｇＱｉｎｈａｉ...     

有限群的c-半置换子群

研究了有限群的结构问题,利用子群c-半置换和完全c-半置换的定义和性质,通过对有限群sylow子群的2-极大子群的研究,获得了有限群幂零、p-幂零的充分条件和另外两个决定群结构的充要条件.     

2-极大子群的$F_s$-拟正规性

$F$是一个群系. $G$的子群$H$在$G$中称为$F_s$-拟正规的,如果存在$G$的正规子群$T$,使得$HT$在$G$中是$s$-置换的并且$(H\cap T)H_G/H_G$包含在$G/H_G$的$F$超中心$Z^F_\infty(G/H_G)$中.本文利用$F_s$-拟正规子群研究了有限群的结构.获得了某些新的结果.     

Finite groups with seminormal Schmidt subgroups

A non-nilpotent finite group whose proper subgroups are all nilpotent is called a Schmidt group. A subgroup A is said to be seminormal in a group G if there exists a subgroup B such that G = AB and AB₁ is a proper subgroup of G, for every proper subgroup B₁ of B. Groups that contain seminormal Schmidt subgroups of even order are considered. In particular, we prove that a finite group is solvable if all Schmidt {2, 3}-subgroups and all 5-closed {2, 5}-Schmidt subgroups of the group are seminormal; the classification of finite groups is not used in so doing. Examples of groups are furnished which show that no one of the requirements imposed on the groups is unnecessary. Supported by BelFBR grant Nos. F05-341 and F06MS-017. __________ Translated from Algebra i Logika, Vol. 46, No. 4, pp. 448–458, July–August, 2007.     

子群次正规性对有限群可解性的影响

考查了次正规子群对有限群结构的影响,得到有限群可解的若干充分条件和超可解的一个充分条件.     

有限可解群的若干充分条件

运用有限群的弱c-正规性,得到了有限可解群的若干充分条件,推广了相关文献的一些结果.