子群交换度对群结构的影响

本文研究了子群交换度与有限群结构的关系.利用子群交换度的数量性质,获得了循环群和2n阶的广义四元数群的一个新刻画.     

特定阶的子群都同构且交换的有限p-群

研究某些子群同构的有限p-群是很有趣的.例如,Hermann和Mann都曾研究过极大子群都同构的有限p-群,但这类群的结构非常复杂,到现在人们都没能给出其分类.研究了特定阶的子群都同构且交换的有限p-群,并给出其分类.     

有限群的CAP-拟正规子群（英文）

有限群G的一个子群A称为G的广义CAP-子群,如果对于任一G-主因子H/K,要么A避免H/K,要么下述成立:(1)如果H/K非交换,那么(A∩H)K/K是H/K的一个Hall子群;(2)如果H/K是一个p-群,那么|G:N_G((A∩H)K)|是一个p-数.G的一个子群H称为在G中是CAP-拟正规的,如果G有一个拟正规子群T和一个广义CAP-子群A满足HT在G中是S-拟正规的并且H∩T≤A≤H.本文得到了CAP-拟正规子群的一些结果并用它们给出一个有限群属于某个包含超可解群的饱和群系的条件.文章推广了很多最近的结果.     

有限p-群的半p-交换性和正则性

本文定义了所谓“半p-交换p-群”(定义3),例1和例2说明半p-交换性与正则性(定义1)并不等价.但定理1证明了:有限p-群是正则的与它的每个子群(包括自身)的任一商群都是半P-交换的这两个条件等价.这就给出了有限正则P-群的一个充分必要条件.     

弱 c- 可置换子群对有限群结构的影响

Guo和Chen在2010年引入了弱c-可置换子群的概念,并利用这种新的广义置换性质对有限群的结构给出了一些新的刻画.本文推广了上述文献中的两个主要结论,确定了极小子群为弱c-可置换子群的有限群的结构.     

有限群的ss-置换子群

设G为有限群,称G的子群H为ss-置换子群,如果存在G的次正规子群B使得G=HB,且H与B的任意Sylow子群可以交换,即对任意X∈Syl（B）有XH=HX.利用子群的ss-置换性来研究有限群的结构,得到有限群超可解的两个充分条件.     

Sylow子群的极大子群在局部子群中的 π - 拟正规性

有限群 G 的一个子群 H 称为在 G 中 π - 拟正规的, 如果 H 和G的每一个Sylow子群可交换. 自从这一概念被 Kegel 提出后, 许多学者相继研究了某些子群在G中的 π - 拟正规性对有限群结构的影响.该文将上述条件局部化,即在群 G 的Sylow 子群的正规化子中来研究这一性质与有限群结构之间的关系.     

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.     

有限交换环上典型群的Carter子群

令R为有限交换局部环,K为其剩余类域,令|K|=q.本文研究了R上辛群Sp2nR和正交群O2nR的Carter子群的存在性及结构,并给出R上正交群O2nR在q≡-1（mod 4）情况下的Sylow 2-子群的正确描述.     

所有子群皆交换或正规的有限群

本文研究所有子群皆交换或正规的有限群. 我们获得了非幂零的情形的一个特征刻画, 也给出了幂零情形的一些性质.     

Subgroup commutativity degrees of finite groups

In this paper we introduce and study the concept of subgroup commutativity degree of a finite group G. This quantity measures the probability of two random subgroups of G commuting. Explicit formulas are obtained for some particular classes of finite groups.     

A note on subgroup commutativity degrees of finite groups

In this note we give some new results concerning the subgroup commutativity degree of a finite group G. These are obtained by considering the minimum of subgroup commutativity degrees of all sections of G.     

A Generalization of Commutativity Degree of Finite Groups

The commutativity degree of a finite group is the probability that two arbitrarily chosen group elements commute. This notion has been generalized in a number of ways. The object of this article is to study yet another generalization of the same notion, which further extends some of the existing generalizations.     

Commutativity of rings and near-rings with generalized derivations

In this paper we prove some theorems of commutativity for prime rings or 3-prime near-rings with a suitably constrained generalized derivation. As a consequence of the results obtained, we prove several commutativity theorems for 3-prime near-rings and prime rings. Also, we prove some other results about the generalized derivation either on a near-ring or on a ring.     

On the multiple exterior degree of finite groups

Recently the first two authors have introduced a group invariant, called exterior degree, which is related to the number of elements x and y of a finite group G such that xΛy = 1 in the exterior square GΛG of G. Research on this topic gives some relations between this concept, the Schur multiplier and the capability of a finite group. In the present paper, we will generalize the concept of exterior degree of groups and we will introduce the multiple exterior degree of finite groups. Among other results, we will obtain some relations between the multiple exterior degree, multiple commutativity degree and capability of finite groups.     

On commutativity of near-rings with generalized derivations

In this paper we prove some theorems of commutativity for near-rings with generalized derivations. As a consequence of the results obtained, we generalize some published results. Also, we give some examples to show that some conditions in some results obtained are not redundant.     

Strong subgroup commutativity degree and some recent problems on the commuting probabilities of elements and subgroups

We show some results on the probability that a randomly picked pair (H, K) of subgroups of a finite group G satisfies [H, K] = 1. This notion of probability is related with the subgroup S-commutativity degree in [D.E. Otera and F.G. Russo, Subgroup S-commutativity degree of finite groups, Bull. Belgian Math. Soc. 19 (2012), 373–382]. Numerical inequalities will be illustrated, in order to find connections with the subgroup commutativity degree and with the commutativity degree of G. On the other hand, the literature has known a significant growth in the last years and so we discuss a series of new open problems, which are arousing interest in the area. The presence of a large bibliography allows us to have a detailed overview of the status of knowledge on the topic.     

Dimension polynomials in the generalized difference case

The aim of this paper is to generalize results on dimension polynomials of difference modules over difference rings for a wider class of rings of difference operators. We introduce the notion of quasi-commutativity, which generalizes the notion of commutativity and enables one to consider wider classes of monoids and groups of endomorphisms. Some properties of quasi-commutative monoids and groups are established; these properties allow us to apply some methods that are almost similar to the ones used in working with free commutative monoids and groups. Also we prove the theorem of existence of the dimension polynomial of generalized difference modules in the cases where the submonoid of endomorphisms is free quasi-commutative. Also the existence of its analog for the case of a direct product of a free quasi-commutative monoid and a finite cyclic group is established.     

Finite rings in which commutativity is transitive

A ring is called commutative transitive if commutativity is a transitive relation on its nonzero elements. Likewise, it is weakly commutative transitive (wCT) if commutativity is a transitive relation on its noncentral elements. The main topic of this paper is to describe the structure of finite wCT rings. It is shown that every such ring is a direct sum of an indecomposable noncommutative wCT ring of prime power order, and a commutative ring. Furthermore, finite indecomposable wCT rings are either two-by-two matrices over fields, local rings, or basic rings with two maximal ideals. We characterize finite local rings as generalized skew polynomial rings over coefficient Galois rings; the associated automorphisms of the Galois ring give rise to a signature of the local ring. These are then used to further describe the structure of finite local and wCT basic rings.