Finite Groups with Subnormal Schmidt Subgroups

We study finite groups some of whose Schmidt subgroups (the minimal nonnilpotent subgroups) are subnormal.     

Indices of Maximal Subgroups of Finite Soluble Groups

We look at the structure of a soluble group G depending on the value of a function m(G)= max m _p G), where m _p(G)=max{log_p|G:M| | M< G, |G:M|=p ^a}, p (G). Theorem 1 states that for a soluble group G, (1) r(G/ (G))= m(G); (2) d(G/ (G)) 1+ (m(G)) 3+m(G); (3) l _p(G) 1+t, where 2^t-1<m _p(G) 2^t. Here, (G) is the Frattini subgroup of G, and r(G), d(G), and l _p(G) are, respectively, the principal rank, the derived length, and the p-length of G. The maximum of derived lengths of completely reducible soluble subgroups of a general linear group GL(n,F) of degree n, where F is a field, is denoted by (n). The function m(G) allows us to establish the existence of a new class of conjugate subgroups in soluble groups. Namely, Theorem 2 maintains that for any natural k, every soluble group G contains a subgroup K possessing the following properties: (1) m(K); k; (2) if T and H are subgroups of G such that K T <_max <_max H G then |H:T|=p ^t for some prime p and for t>k. Moreover, every two subgroups of G enjoying (1) and (2) are mutually conjugate.     

Maximal and Sylow Subgroups of Solvable Finite Groups

The structure of finite solvable groups in which any Sylow subgroup is the product of two cyclic subgroups is studied. In particular, it is proved that the nilpotent length of such a group is no greater than 4. It is also proved that the nilpotent length of a finite solvable group in which the index of any maximal subgroup is either a prime or the square of a prime or the cube of a prime does not exceed 5.     

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.     

有限群的弱S-拟正规嵌入子群

群G的子群H称为在G中S-拟正规嵌入的,如果对于任意的素数p||H|,H的Sylow p-子群也是G的某个S-拟正规子群的Sylow p-子群.称群G的子群H在G中弱S-拟正规嵌入,如果存在群G的正规子群T,使得HT■G且H∩T在G中是S-拟正规嵌入的.研究了弱S-拟正规嵌入子群的性质,给出了某些群类的新的特征,并推广了一些已知的结论.     

n-极大子群为C-正规的有限群

首先,利用Hall子群的C－正规性,得到了有限群成为。可解群的一个充分条付,推广了Schur-Zassenhaus定理;其次,考查了 n－极大子群或其 2阶及 4阶循环子群为 C-正规对有限群结构的影响．     

On SQ-Supplemented Subgroups of Finite Groups

Let G be a finite group and H ≤ G. The subgroup H is called: S-permutable in G if HP = PH for all Sylow subgroups P of G; S-permutably embedded in G if each Sylow subgroup of H is also a Sylow subgroup of some S-permutable subgroup of G.

Let H be a subgroup of a group G. Then we say that H is SQ-supplemented in G if G has a subgroup T and an S-permutably embedded subgroup C ≤ H such that HT = G and T ∩ H ≤ C.

We study the structure of G under the assumption that some subgroups of G are SQ-supplemented in G. Some known results are generalized.     

有限群的X-s-置换子群

Let X be a nonempty subset of a group G.A subgroup H of G is said to be X-s-permutable in G if there exists an element x ∈ X such that HPx = PxH for every Sylow subgroup P of G.In this paper,some new results are given under the assumption that some suited subgroups of G are X-s-permutable in G.     

有限群的超可解性

本文利用子群格的模的性质对有限群进行研究 ,得到了有限超可解群的若干充分条件 ,推广了许多已有结果 .     

群与代数的表示理论（迎接ICM2002特约文章）

有限群模表示论,代数群与量子群,代数表示论,同调代数与代数K-理论是当前国际数学研究的前沿重点领域,在20世纪得到巨大发展,它们之间的相互交叉融合与综合统一在数学发展中具有鲜明特色。本文简要概述上述研究领域的发展历史与最新研究进展。     

Finite Groups with S-permutable n-maximal Subgroups

Let G be a finite group and M _n (G) be the set of n-maximal subgroups of G, where n is an arbitrary given positive integer. Suppose that M _n (G) contains a nonidentity member and all members in M _n (G) are S-permutable in G. Then any of of the following conditions guarantees the supersolvability of G: (1) M _n (G) contains a nonidentity member whose order is not a prime; (2) all nonidentity members in M _n (G) are of prime order, and all cyclic members in M _n?1(G) of order 4 are S-permutable in G.     

On Nearly SS-Embedded Subgroups of Finite Groups

Let H be a subgroup of a finite group G. H is nearly SS-embedded in G if there exists an S-quasinormal subgroup K of G, such that HK is S-quasinormal in G and H ∩ K ≤ HseG, where HseG is the subgroup of H, generated by all those subgroups of H which are S-quasinormally embedded in G. In this paper, the authors investigate the influence of nearly SS-embedded subgroups on the structure of finite groups.     

有限交换环上典型群的Sylow子群

令Ｒ为有限交换局部环，Ｍ表其唯一的极大理想，ｋ表其剩余类域．本文定出了Ｒ上的一般线性群ＧＬｎＲ，辛群Ｓｐ２ｎＲ及双曲正交群Ｏ２ｎＲ的Ｓｙｌｏｗ子群．一般讲，若ｃｈａｒｘ＝ｐ，上述三类典型群的Ｓｙｌｏｗ ｐ－子群分别同构于由某些特殊形式的矩阵生成的子群；若ｃｈａｒｋ≠ｐ，上述三类典型群的Ｓｙｌｏｗｐ－子群分别同构于一循环群或半二面体群与若干Ｚｐ型循环群的圈积。     

有限群的s-条件置换子群

如果对群G的任意Sylow子群T,存在一个元素x∈G,使得HT~x=T~xH,那么称群G的子群H在G中s-条件置换．利用s-条件置换子群给出了一些群的性质和结构．     

二次极大子群中2阶及4阶循环子群拟正规的有限群

本文讨论２阶及４阶循环子群对群结构的影响．主要结果是下述定理：如果有限群Ｇ满足标题的条件，那么下列情形之一成立：（１）Ｇ有正规Ｓｙｌｏｗ ２－子群；（２） Ｇ为 ２－幂零；（３） Ｇ ≌ Ｓ４；（４） Ｇ＝ＰＱ，其中 Ｐ为阶 ２４广义四元数群， Ｑ为 ３阶循环群；（５） Ｇ ≌ Ａ５或 ＳＬ（２，５）．     

On c-Normal Subgroups of Finite Groups

Let G be a finite group. We fix in every noncyclic Sylow subgroup P of G some subgroup D satisfying 1 < |D| < |P| and study the structure of G under the assumption that all subgroups H of P with |H| = |D| are c-normal in G.     

A Note on c-Supplemented Subgroups of Finite Groups

A subgroup H of a finite group G is said to be c-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K is contained in H _G , where H _G is the largest normal subgroup of G contained in H. In this article, we prove that G is solvable if every subgroup of prime odd order of G is c-supplemented in G. Moreover, we prove that G is solvable if and only if every Sylow subgroup of odd order of G is c-supplemented in G. These results improve and extend the classical results of Hall's articles of (1937) and the recent results of Ballester-Bolinches and Guo's article of (1999), Ballester-Bolinches et al.'s article of (2000), and Asaad and Ramadan's article of (2008).     

具有若干$c_p$-可补子群的有限群

设$H$为有限群$G$的子群且$p$为整除群$G$的阶的素因子. 我们称$H$在$G$中为$c_p$-可补的,如果$G$中存在$H$的包含$H_G$的补子群$T$使得$H\cap T/H_G$为$p''$-群, 其中$H_G$为$H$在$G$中的核. 群$G$ 称$CS_p$-群, 如果$G$的所有$p$-子群都在$G$中$c_p$-可补. 本文,我们刻画具有若干$c_p$-可补$p$-子群的有限群的$p$-可解性和$p$-超可解性. 此外,我们给出$p$-可解群为$CS_p$-群的若干等价条件. 最后, 我们给出两个$CS_p$-群的直积为$CS_p$-群的判别准则. 我们的结果推广了近期的若干结论.     

弱SS-半置换性极小子群对有限群超可解性的影响

引入弱SS-半置换子群的概念,介绍了弱SS-半置换子群的性质,结合有限群G的极小于群的弱SS-半置换性,并结合C-正规性来讨论有限群的超可解性及幂零性,得到了有限群超可解及幂零的若干充分或充要条件,同时推广了某些著名结果.