Connected cubic s-arc-regular Cayley graphs of finite nonabelian simple groups

A graph is said to be s-arc-regular if its full automorphism group acts regularly on the set of its s-arcs. In this paper, we investigate connected cubic s-arc-regular Cayley graphs of finite nonabelian simple groups. Two suffcient and necessary conditions for such graphs to be 1- or 2-arc-regular are given and based on the conditions, several infinite families of 1-or 2-arc-regular cubic Cayley graphs of alternating groups are constructed.

Finite p-groups All of Whose Minimal Nonabelian Subgroups are Nonmetacyclic of Order p^3

Assume p is an odd prime. We investigate finite p-groups all of whose minimal nonabelian subgroups are of order p^3. Let P1-groups denote the p-groups all of whose minimal nonabelian subgroups are nonme tacyclic of order p^3. In this paper, the P1-groups are classified, and as a by-product, we prove the Hughes' conjecture is true for the P1-groups.

Nonabelian omni-Lie algebroids

In this paper,we study the structure of nonabelian omni-Lie algebroids.Firstly,taking Lie algebroid(E,[·,·]_(E,ρE))as the starting point,a nonabelian omni-Lie algebroid is defined on direct sum bundle DE⊕JE,where DE and JE are,respectively,the gauge Lie algebroid and the jet bundle of vector bundle E,and study its properties.Furthermore,it is concluded that the nonabelian omni-Lie algebroid is a trivial deformation of the omni-Lie algebroid,and the nonabelian omni-Lie algebroid is a matched pair of Leibniz algebroids.

关于内有限的无限单群

主要对内有限的无限单群进行了研究,得到了:若非交换的内有限群含有对合,则非单群;而且内有限的无限单群是内可解的.在此基础上得到了:内有限的无限单群分别为内幂零、内交换、内循环的充分必要条件.

p-群中不同阶元素的可交换性研究

对任意两个不同阶元素可交换的p-群进行了研究,得到:(1)当G为方次数为p的p-群时,G中任意两个不同阶元素均可交换;当G为方次数大于p的p-群时,G中任意两个不同阶元素均可交换充分必要条件是Ωe-1(G)≤Z(G)。(2)设G为非交换的p-群,若G中任意两个非正规的子群均有相同的阶,则G中任意两个不同阶元素均可交换。

一类非交换p-群的正规子群的补

针对有限群G的正规子群的补子群的存在性问题,Schur-Zassenhaus定理从群的阶的角度给出了补子群存在的充分条件以及所有补子群之间的关系;在群作用的环境下Frattini论断给出了正规子群的补子群存在的充分条件。在此基础上,计算了一类非交换p^(n+1)(p≠2,n≥2)阶群的正规子群的补子群的结构,做为应用,给出了p^(3)阶非交换p-群的补子群存在的充分条件。

PN群与半完全群的关系

本文引入纯非交换群的概念,利用半完全群与自同构群的密切关系,得到了纯非交换群与半完全群之间的关系及性质。

PN群与半完全群的关系及性质

引入纯非交换群的概念,利用自同构群与完全群的密切关系,得到了纯非交换群与半完全群两者之间的关系.

平展φ-模的上同调刻画

利用非交换循环上同调理论刻画了平展φ-模的同构类.确切地说,设R是一个环,给出了上同调集合H1(〈φ〉,GLd(R))和R上的自由平展φ-模的同构类之间的一一对应.此外,还给出了一个类似于希尔伯特定理90的结论.这里的方法还可以用来刻画(φ,Γ)-模的同构类.

造27阶非交换群的特征标表

对27阶非交换群在共轭分类的基础上给出其特征标表.

非交换的非平凡子群均有唯一非平凡特征子群的有限p群

本文得到了以下结果:设G为非内交换的有限非交换p-群,本文给出了非交换的非平凡子群均有唯一非平凡特征子群的群G结构.

Some Simple Groups Which are Determined by Their Orders and Degree-Patterns

Let G be a finite group and Irr(G)the set of all irreducible complex characters of G.Let cd(G)be the set of all irreducible complex character degrees of G and denote byρ(G)the set of all primes which divide some character degree of G.The character-prime graphΓ(G)associated to G is a simple undirected graph whose vertex set isρ(G)and there is an edge between two distinct primes p and q if and only if the product p q divides some character degree of G.We show that the finite nonabelian simple groups A_(7),J_(1),J_(3),J_(4),L_(3)(3)and U_(3)(4)are uniquely determined by their degree-patterns and orders.

Finite p-groups whose nonnormal subgroups are metacyclic

For an odd prime p,we give a criterion for finite p-groups whose nonnormal subgroups are metacyclic,and based on the criterion,the p-groups whose nonnormal subgroups are metacyclic are classified up to isomorphism.This solves a problem proposed by Berkovich.

幂零Hall π-子群的存在性

本文首先将Ｈａｌ定理推广为：设Ｎ为Ｇ的正规子群，若Ｎ为Ｅｎπ群，Ｇ／Ｎ为Ｄπ群，则Ｇ为Ｄπ群．在此基础上得到了群Ｇ为Ｅｎπ群的充要条件为：（１）Ｇ存在正规子群Ｎ，满足Ｎ及Ｇ／Ｎ为Ｅｎπ群；（２）对任意ｐ∈π，任意ｑ∈π｛ｐ｝及任意ｐ元素ｘ，ＣＧ（ｘ）含Ｇ的Ｓｙｌｏｗｑ子群．另外，我们对非Ａｂｌｅ单群的情形也进行了一些讨论．

On B6-and B7-groups

A finite group G is said to be a Bn-group if any n-element subset A={a1,a2,...,an}of G satisfies∣∣A^2∣∣=|{aiaj|1≤i,j≤n}|≤n(n+1)/2.In this paper,the characterizations of the B6-and B7-groups are given.

Finite 2-Groups Whose Number of Subgroups of Each Order are at Most 2^(4)

Assume G is a group of order 2^(n),n≥5.Let s_(k)(G)denote the number of subgroups of order 2^(k) of G.We classify finite 2-groups G with s k(G)≤2^(4),where 1≤k≤n.

非交换群上的2度有向Cayley图

称群G的有向Cayley图X=Cay(G,S)是正规的,如果群G的右正则表示R(G)在图X的全自同构群Aut(X)中正规.给出了非交换群的2度有向Cayley图是正规的充分条件.另外,利用自同构群和查圈的方法决定了广义四元数群上的2度有向Cayley图的正规性.

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.

A Characterization of the Smallest Suzuki 2-Group

Let G be a finite nonabelian group which has no abelian maximal subgroups and satisfies that any two non-commutative elements generate a maximal subgroup. Then G is isomorphic to the smallest Suzuki 2-group of order 64.

On Symmetric Cayley Graphs of Valency Eleven

A Cayley graph Г=Cay(G,S)is said to be normal if G is normal in Aut Г.In this paper,we investigate the normality problem of the connected 11-valent symmetric Cayley graphs Г of finite nonabelian simple groups G,where the vertex stabilizer A_(u) is soluble for A=Aut Г and v∈∨Г.We prove that either Г is normal or G=A_(5),A_(10),A_(54),A_(274),A_(549) or A_(1099).Further,11-valent symmetric nonnormal Cayley graphs of As,A54 and A274 are constructed.This provides some more examples of nonnormal 11-valent symmetric Cayley graphs of finite nonabelian simple groups after the first graph of this kind(of valency 11)was constructed by Fang,Ma and Wang in 2011.