l-群的凸l-子群格的极小条件

设G是L-群,C(G)是G的凸l-子群格．称C(G)满足极小条件,如果C(G)中每个元均包含一个原子元．本文将C(G)的链条件(见文[1])推广到极小条件,主要结果是：C(G)满足极小条件且C(G)中每个原子元均是G的基数直和项当且仅当∑rλ∈∪∈Пλ∈ARλ(其中每个Rλ≌实数加群R的某个子群)。     

关于格序群的极大素子群

研究了格序群的极大素子群的性质以及由素子群根系确定的几种格序群类的结构。     

弱奇异l-群的结构

引入弱奇异元及弱奇异l-群的概念，通过建立弱奇异元及弱奇异l-群的刻划，研究了一般弱奇异l-群的性质及相关的结构，部分地改进了有关奇异l-群已有的结果．     

l-群的凸-子群格的极小条件

设G是l-群,C(G)是G的凸l-子群格.称C(G)满足极小条件,如果C(G)中每个元均包台一个原子元.本文将C(G)的链条件(见文[1])推广到极小条件,主要结果是:C(G)满足极小条件且C(G)中每个原子元均是G的基数直和项当且仅当∑λ∈ΛRλ(∩)G(∩)Пλ∈ΛRλ(其中每个Rλ≌实数加群R的某个子群).     

极小非PNN-群

有限群G称作PNN-群,如果G的每一个极小子群X要么正规于G,要么是自正规化的.本文将讨论PNN-群的性质,并给出极小非PNN-群,也就是本身不是PNN-群,每个真子群都是PNN-群的有限群的完全分类.     

l-群的极大下子群

在非正规值的条件下,给出l-群极大下子群的唯一表达形式,由此推广[1]中的一个结果.与此同时,建立了正规值l-群,特殊值l-群及有限值l-群之间等价的一个充要条件.     

特殊值l-群的l-同态象及l-扩张性

本文研究了特殊值 l-群的一类特殊的 l-同态象及 l-扩张闭性 .在非超 Archimedean的条件下 ,证明了 l-群 G是特殊值的当且仅当对于 G的每个闭 l-理想 K,G/ K与 K均是特殊值的 .     

非可换的奇异l-群

通过建立奇异元以及奇异l-群的刻划，研究了一般非可换的奇异l-群的性质及相关的结构．     

二次极大子群中2阶及4阶循环子群拟中心的有限群

本文讨论2阶及4阶循环子群对群结构的影响．证明二次极大子群中2阶及4阶循环子群拟中心的有限群G同构于下列群之一：(1)G为2-闭群；(2)G为2-幂零群；(3)G≌S，；(4)G=PQ．其中P为2^4阶广义四元数群，Q≌C3；(5)G≌A5或SL(2，5)．     

极小非MSSP-群的分类

本文研究了一类有限极小非∑-群.利用群G的每个Sylow子群的极小子群的s-半置换性对G的结构影响,获得了一类新的极小非∑-群的分类,推广了极小非Σ-群部分研究成果.     

关于有限群的Pronormal极小子群

利用有限群G的pronormal极小子群和Sylow子群正规化子中的素数阶弱左Engel元素得到了G成为p-幂零群、幂零群和超可解群的一些充分条件,这些结果推广了已知结论。     

Finite p-groups All of Whose Minimal Nonabelian Subgroups are Nonmetacyclic of Order p3

Assume p is an odd prime. We investigate finite p-groups all of whose minimal nonabelian subgroups are of order p³. Let P₁-groups denote the p-groups all of whose minimal nonabelian subgroups are nonmetacyclic of order p³. In this paper, the P₁-groups are classified, and as a by-product, we prove the Hughes' conjecture is true for the P₁-groups.     

Quasi-Purifiable Subgroups and Minimal Direct Summands

Let G be an arbitrary Abelian group. A subgroup A of G is said to be quasi-purifiable in G if there exists a pure subgroup H of G containing A such that A is almost-dense in H and H/A is torsion. Such a subgroup H is called a “quasi-pure hull” of A in G. We prove that if G is an Abelian group whose maximal torsion subgroup is torsion-complete, then all subgroups A are quasi-purifiable in G and all maximal quasi-pure hulls of A are isomorphic. Every subgroup A of a torsion-complete p-primary group G is contained in a minimal direct summand of G that is a minimal pure torsion-complete subgroup containing A. An Abelian group G is said to be an “ADE decomposable group” if there exist an ADE subgroup K of G and a subgroup T′ of T(G) such that G = K⊕ T′. An Abelian group whose maximal torsion subgroup is torsion-complete is ADE decomposable. Hence direct products of cyclic groups are ADE decomposable groups.     

Finite Groups with Some Subgroups of Prime Power Order S-Quasinormally Embedded

Abstract

A subgroup of a group G is said to be S-quasinormal in G if it permutes with every Sylow subgroup of G. A subgroup H of a group G is said to be S-quasinormally embedded in G if every Sylow subgroup of H is a Sylow subgroup of some S-quasinormal subgroup of G. In this paper we examine the structure of a finite group G under the assumption that certain abelian subgroups of prime power order are S-quasinormally embedded in G. Our results improve and extend recent results of Ramadan [Ramadan, M. (2001). The influence of S-quasinormality of some subgroups of prime power order on the structure of finite groups. Arch. Math. 77:143–148].     

Subgroups of totally imprimitive permutation groups

In this work the study of subgroups of totally imprimitive permutation groups is continued. Some sufficient conditions for the existence of minimal non-FC-subgroups are given. These results depend mainly on the properties of self-normalizing subgroups, which are closed in the topology of point-wise convergence. Finally, new properties of a totally imprimitive permutation p-group satisfying the cyclic-block property are given which might be a suitable example to look for minimal non-FC-subgroups.     

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 Maximal and Minimal Subgroups of Sylow Subgroups of Finite Groups. II

Abstract

A subgroup H of G is said to be c-normal in G if there exists a normal subgroup N of G such that HN = G and H ∩ N ≤ H _G  = Core(H). We extend the study on the structure of a finite group under the assumption that all maximal or minimal subgroups of the Sylow subgroups of the generalized Fitting subgroup of some normal subgroup of G are c-normal in G. The main theorems we proved in this paper are:

Theorem Let ? be a saturated formation containing 𝒰. Suppose that G is a group with a normal subgroup H such that G/H ∈ ?. If all maximal subgroups of any Sylow subgroup of F*(H) are c-normal in G, then G ∈ ?.

Theorem Let ? be a saturated formation containing 𝒰. Suppose that G is a group with a normal subgroup H such that G/H ∈ ?. If all minimal subgroups and all cyclic subgroups of F*(H) are c-normal in G, then G ∈ ?.