利用极大子群的θ-子群偶判定有限群的可解性

主要从两类特殊的极大子群出发,利用其极大θ-子群偶给出有限群可解的一些充要条件.     

关于有限群极大子群的强θ-完备

本文定义了有限群极大子群的强θ-完备这一概念.通过研究极大子群的强θ-完备对群结构的影响,给出了有限群可解性,超可解性,幂零性的一些新刻画,所得结果改进了以往的定理.     

关于有限群极大子群的强θ-完备

本文定义了有限群极大子群的强θ-完备这一概念．通过研究极大子群的强θ-完备对群结构的影响,给出了有限群可解性,超可解性,幂零性的一些新刻画,所得结果改进了以往的定理．     

极大子群的θ-子群及可解性

本文研究有限群极大子群的θ－子各对可解性的影响,建立了可解性的一些新条件。     

域上 F4型 Chevalley 群中含单项子群的极大子群

对特征不为2的任意域上F4型Chevalley群,构造了一类真包含单项子群的子群,从而否定回答了单项子群的极大性问题.同时证明了所构造的群恰为极大子群.     

关于一类极大子群的θ-偶

对于有限群G的极大子群M，令β（G：M）表示整除│G：M│的素因子个数，β（G）表示所有β（G；M）中的最大数．令μ（G）为使得β（G：M）=β（G）的极大子群的集合．通过对这一类极大子群的θ-偶赋予一定条件，得到了判断群G可解、超可解的新结果．     

域上Cl型Chevalley群中含单项子群的极大子群

对特征不为2的任意域上Cl型Chevalley群.构造了一类真包含单项子群的子群,从而否定回答了单项子群的极大性问题,同时证明了所构造的群恰为极大子群.     

有限群极大子群的s-θ-完备与π-可解性

定义一些极大子群的集合,通过研究极大子群的s-θ-完备对群结构的影响,给出有限群为π-可解的一些新刻画.     

子群的θ-偶和群的结构

研究极大子群和2-极大子群的θ-偶对群结构的影响.设G是有限群,本文得到了:如果G的每一个极大子群M都有极大θ-偶(C,D),使MC=G且C/D是2-闭的,那么G可解;如果G的每一个2-极大子群H都有θ-偶(C,D),使C/D幂零且G=HC,那么G是幂零.     

域上F_4型Chevalley群中含单项子群的极大子群

对特征不为 ２的任意域上 Ｆ_４型 Ｃｈｅｖａｌｌｅｙ群,构造了一类真包含单项子群的子群,从而否定回答 了单项子群的极大性问题,同时证明了所构造的群恰为极大子群     

On the poset of conjugacy classes of subgroups of finite p-groups

This paper is a continuation of the paper “On the Deskins completions, theta completions and theta pairs for maximal subgroups ”.In the former paper, Zhao Yaoqing introduced the concept of θ-completions associated to a maximal subgroup of a finite group. The concept offers a convenience for us to study the completions introduced by Deskins and gives us a way to reveal the relationship between the concepts of completions and θ-pairs, the latter concept is introduced by Mukherjee and Bhattacharya. The present paper is devoted to discussing the π-solvability, π-supersolvability and π-nilpotency of a finite group by using the θ-completions. Moreover, a new proof on the Deskins conjecture concerning the supersolvability is included.     

On the desking completions,theta completions and theta pairs for maximal subgroups *

Given a maximal subgroup M of a finite group G,a θ completion of M in G is any subgroup C such that C is not contained in M while M_G , the core of M in G, is contained in C and C/M_G has no propor normal subgroup of G/M_G . By using this concept we can reveal the relationship between the concepts of completions and θ-pairs introduced respectively by Deskins, Mukherjee and Bhattacharya. The concept of maximal θ-completions offers a convenience for us to study the Deskins completions in inductions. We obtain in this paper several results which imply a group to be solvable, supersolvable and nilpotent.     

On theta pairs for maximal subgroups ∗

For a maximal subgroup M of a finite group G, a θ -pair is any pair of subgroups (CD) of G such that (i) DjG, D>C, (ii) (M, C=G, <M,D> = M and (iii) C/D has no proper normal subgroup of G/D. A natural partial ordering is defined on the family of 0 -pairs. We study the further properties of the maximal 0 -pairs of M and obtain several results on 0 -pairs which imply G to be n -solvable, 7t - supersolvable and π - nilpotent     

p-Fitting子群的CAP-嵌入子群

群G的子群H称为G的CAP-嵌入子群,如果对于|H|的每个素因子p,存在G的某个CAP-子群K,使得H的某个Sylow p-子群也是K的一个Sylowp-子群.本文通过假定G的p-Fitting子群F_p(G)的某个Sylow p-子群的每个极大子群是G的CAP-嵌入子群,得到一些新的结果.     

具有弱$s$-置换嵌入子群的有限群

Let P be a Sylow p-subgroup of a group G with the smallest generator number d,where p is a prime.Denote by M_d(P) = {P_1,P_(2,...,)P_d} a set of maximal subgroups of P such that φ(P) = ∩_(n=1)~dP_n.In this paper,we investigate the structure of a finite group G under the assumption that the maximal subgroups in M_d(P) are weakly s-permutably embedded in G,some interesting results are obtained which generalize some recent results.Finally,we give some further results in terms of weakly s-permutably embedded subgroups.     

半次覆盖远离子群和有限群的可解性

本文定义了有限群的半次覆盖远离子群概念,研究了半次覆盖远离子群和有限群的可解性问题.利用某些半次覆盖远离子群刻划了有限群的可解性,得到了若所有的sylow子群(或极大子群)半次覆盖远离则群可解,推广了文献[6]中的结果.     

On p-nilpotency and minimal subgroups of finite groups

We call a subgroup H of a finite group G c-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K ≤ core(H). In this paper it is proved that a finite group G is p-nilpotent if G is S4-free and every minimal subgroup of P n GN is c-supplemented in NG(P), and when p = 2 P is quaternion-free, where p is the smallest prime number dividing the order of G, P a Sylow p-subgroup of G. As some applications of this result, some known results are generalized.     

Frattini子群循环的有限$p$-群中的非交换集和极大Abel子群

设G是一个群,X是G的一个子集,若对于任意x,y∈X且x≠y,都有xy≠yx,则称X是G的一个非交换集.进一步,如果对于G中的任意其它非交换子集Y,都有|X|≥|Y|,那么称X是G的一个极大非交换集.文中确定了Frattini子群循环的有限p-群中极大非交换集和极大Abel子群的势.     

On finite 2-groups all of whose subgroups are mutually isomorphic

In this paper we investigate the title groups which we call isomaximal. We give the list of all isomaximal 2-groups with abelian maximal subgroups. Further, we prove some properties of isomaximal 2-groups with nonabelian maximal subgroups. After that, we investigate the structure of isomaximal groups of order less than 64. Finally, in Theorem 14. we show that the minimal nonmetacyclic group of order 32 possesses a unique isomaximal extension of order 64. This work was supported by Ministry of Science, Education and Sports of Republic of Croatia (Grant No. 036-0000000-3223)