阿贝尔群被超-(循环或有限)群的可裂扩张(Ⅰ)

设F是由f(p)所局部定义的可解群系,G∈F,A是ZG-模.我们称A的一个p-主因子U/V在G中是F-中心的,如果G/CG(U/V)∈f(p).否则称U/V在G中是非中心的.本文证明了:设G是超-(有限或循环)的局部可解群,A是Artinian ZG-模且所有的不可约ZG-因子都是有限的;F为由f(p)所局部定义的局部可解群系,且对任意的p∈π,f(p)≠φ,f(∞) f(p).如果G∈F,且A的所有不可约ZG-因子在G中均是F-非中心的,则A被G的扩张在A上共轭可裂..     

On Symmetric Cayley Graphs of Valency Eleven

A Cayley graph Γ=Cay(G,S)is said to be normal if G is normal in Aut Γ.In this paper,we investigate the normality problem of the connected 11-valent symmetric Cayley graphs Γ of finite nonabelian simple groups G,where the vertex stabilizer Av is soluble for A=Aut Γ and v ∈ VΓ.We prove that either Γ is normal or G=A5,A10,A54,A274,A549 or A1099.Further,11-valent symmetric nonnormal Cayley graphs of A5,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.     

The deformation theory of representations of fundamental groups of compact Kähler manifolds

Let Γ be the fundamental group of a compact Kähler manifold M and let G be a real algebraic Lie group. Let ?(Γ, G) denote the variety of representations Γ → G. Under various conditions on ρ ∈ ?(Γ, G) it is shown that there exists a neighborhood of ρ in ?(Γ, G) which is analytically equivalent to a cone defined by homogeneous quadratic equations. Furthermore this cone may be identified with the quadratic cone in the space \(Z^1 (\Gamma ,g_{Ad\rho } )\) of Lie algebra-valued l-cocycles on Γ comprising cocyclesu such that the cohomology class of the cup/Lie product square [u, u] is zero in \(H^2 (\Gamma ,g_{Ad\rho } )\) . We prove that ?(Γ, G) is quadratic at ρ if either (i) G is compact, (ii) ρ is the monodromy of a variation of Hodge structure over M, or (iii) G is the group of automorphisms of a Hermitian symmetric space X and the associated flat X-bundle over M possesses a holomorphic section. Examples are given where singularities of ?(Γ, G) are not quadratic, and are quadratic but not reduced. These results can be applied to construct deformations of discrete subgroups of Lie groups.     

有限群的最大子群的性质对群结构的影响

有限群G的一个子群称为在G中是π-拟正规的若它与G的每一个Sylow-子群是交换的.G的一个子群H称为在G中是c-可补的若存在G的子群N使得G=HN且H∩N≤HG=CoreG(H).本文证明了:设F是一个包含超可解群系u的饱和群系,G有一个正规子群H使得G/H∈F.则G∈F若下列之一成立:(1)H的每个Sylow子群的所有极大子群在G中或者是π-拟正规的或者是c-可补的;(2)F*(H)的每个Sylow子群的所有极大子群在G中或者是π-拟正规的或者是c-可补的,其中F*(H)是H的广义Fitting子群.此结论统一了一些最近的结果.     

A New Type of Pronormal Subgroups in Chevalley Groups

Let L be a simple Lie algebra with irreducible root system φ having roots of different length,F be a field of characteristic different from 2,G=L(F) be a Chevalley group of type L over F.Denote by φ^1 the set of all long roots in φ.Set G^1=(zr(t);r∈φ^t,t∈F).It is a subgroup of G generated by all the long root subgroups.This paper determines the pronormality of G^1 in G when L is not of type G2.     

The Poincaré Series of Relative Invariants of Finite Pseudo-Reflection Groups

Let F be a field with characteristic 0,V=F~n the n-dimensional vector space over F and let G be a finite pseudo-reflection group which acts on V.Let χ:G→F~* be a 1-dimensional representation of G.In this article we show that X(g)=(detg)~α(0≤α≤r-1),where g∈G and r is the order of g.In addition,we characterize the relation between the relative invariants and the invariants of the group G,and then we use Molien's Theorem of invariants to compute the Poincaré series of relative invariants.     

图中推广的正交因子分解

设G是一个图,具有顶点集合V(G)和边集合E(G).设g和f是定义在V(G)上的整数值函数,使对每个x∈V(G),有g(x)≤f(x).图G的一个(g,f)-因子是G的一个支撑子图H,使对每个x∈V(G),有g(x)≤d_H(x)≤f(x).G的一个(g,f)-因子分解是E(G)的边不相交的(g,g)-因子的一个划分.设F={F-1,F_2,…,F_m}为G的一个因子分解,H是G的一个有mr条边的子图.如果每个F_i恰好与H有r条公共边,1≤i≤m,则称Fr-正交于H.本文证明每个(mg+kr,mf-kr)-图含有一个子图R,使R有(g,f)-因子分解r-正交于任意给定的有kr条边的子图,其中m,k和r为正整数且k相似文献   

强向量拟均衡问题解的存在性(英文)

设X是一个实的Hausdorff拓扑向量空间,Y是一个实的局部凸向量空间,C是Y中的闭凸锥,K（?）X是一个紧子集.F:X×X→Y是一个双向量函数,G:K→2K是一个集合值映射.我们考虑下面的强拟均衡问题:存在x∈G（x）,使得对任意的y∈G（x）,成立F（x,y）∈C.本文证明了当F是半连续时,上述问题解的存在性结论.     

Sylow－正规化子属于群系F的有限群

设Ｆ是可解的，子群闭的，由｛ｆ（Ｐ）｝所局部定义的群系，Ｆｐ是由｛ｆ（ｑ）｝定义的ｐ－局部定义群系．Ｎ为幂零群系．本文证明了：１）设Ｆ满足：任一群属于Ｆ，当且仅当，对每ｐ．其ｐ－Ｓｙｌｏｗ－正规化子属于Ｆｐ．于是“群Ｇ∈Ｎ．Ｆ（幂零由Ｆ的扩张）的充要条件是，对每Ｐ，其ｐ－Ｓｙｌｏｗ－正规化子的Ｆｐ剩余次正规于Ｇ内．２）群Ｇ为超可解的充要条件是，对每ｐ，其ｐ－Ｓｙｌｏｗ－正规化子为ｐ－超可解，且其幂零剩余次正规于Ｇ内．若对每ｐ，群Ｇ的ｐ－Ｓｙｌｏｗ子群无商群与ｐ２－次对称群的ｐ－Ｓｙｌｏｗ子群同构，则称Ｇ为Ｂ－群．３）设Ｇ为Ｂ－群，又群系Ｆ含于σ－Ｓｙｌｏｗ塔群系内．于是①Ｇ∈Ｆ，当且仅当，对每ｐ，Ｇ的ｐ－Ｓｙｌｏｗ－正规化属于Ｆｐ；②Ｇ∈Ｎ·Ｆ，当且仅当，对每ｐ，Ｇ的ｐ－Ｓｙｌｏｗ－正规化子的Ｆｐ剩余在Ｇ内次正规．     

上连通和超连通的三次Bi-Cayley图

Let G be a finite group and let S(possibly, contains the identity element) be a subset of G. The Bi-Cayley graph BC(G, S) is a bipartite graph with vertex set G×{0, 1} and edge set {(g, 0) (sg, 1) : g∈G, s ∈ S}. A graph is said to be super-connected if every minimum vertex cut isolates a vertex. A graph is said to be hyper-connected if every minimum vertex cut creates two components, one of which is an isolated vertex. In this paper, super-connected and/or hyper-connected cubic Bi-Cayley graphs are characterized.     

Fitting Height and Diameter of Character Degree Graphs

Let a graph Γ have bounded Fitting height (i.e., there is a bound on the Fitting heights of those groups whose character degree graph is Γ) and G be any solvable group with character degree graph Γ and Fitting height h(G). We improve Moretò's bound by proving that if no vertex in Γ is adjacent to every other one, then h(G) ≤4, else h(G) ≤6. As a consequence, if a solvable group G has character degree graph with diameter 3, then h(G) ≤4. Moreover, G has at most one non-abelian normal Sylow subgroup in this case.     

<Emphasis Type="Italic">n</Emphasis>-transitivity of bisection groups of a Lie groupoid

The notion of n-transitivity can be carried over from groups of diffeomorphisms on a manifold M to groups of bisections of a Lie groupoid over M. The main theorem states that the n-transitivity is fulfilled for all n ∈ N by an arbitrary group of C^r-bisections of a Lie groupoid Γ of class C^r, where 1 ≤ r ≤ ω, under mild conditions. For instance, the group of all bisections of any Lie groupoid and the group of all Lagrangian bisections of any symplectic groupoid are n-transitive in the sense of this theorem. In particular, if Γ is source connected for any arrow γ ∈ Γ, there is a bisection passing through γ.     

向量空间中拟不变元素的结构

设Ｖ是有限域上Ｆ的向量空间,Ｇ是Ｖ上的线性变换群．本文讨论了Ｖ中拟不变元素的结构·即如果Ｕ是Ｖ中的拟不变元,则存在ｇ∈Ｇ,使得Ｕ∩ｇ（Ｕ）是Ｇ-不变的,或存在ｘ∈ Ｖ＼Ｕ,使得 Ｖ＋是 Ｇ不变的．     

Stabilité de la conjecture de Baum—Connes pour certains produits semi-directs de groupes

Let Γ = G ⋉_r N be the semi-direct product of two locally compact groups, and A a Y-algebra. We build a map from topological K-theory of Γ with coefficients in A to the one for G with coefficients in N ⋉_r A. For Γ a p-adic group equipped with a γ element and N an amenable group, we prove that Γ satisfies Baum-Connes conjecture with coefficients when G does.     

The quasi-isometry classification of rank one lattices

Let X be a symmetric space—other than the hyperbolic plane—of strictly negative sectional curvature. Let G be the isometry group of X. We show that any quasi-isometry between non-uniform lattices in G is equivalent to (the restriction of) a group element of G which commensurates one lattice to the other. This result has the following corollaries:

1. Two non-uniform lattices in G are quasi-isometric if and only if they are commensurable.
2. Let Γ be a finitely generated group which is quasi-isometric to a non-uniform lattice in G. Then Γ is a finite extension of a non-uniform lattice in G.
3. A non-uniform lattice in G is arithmetic if and only if it has infinite index in its quasi-isometry group.

     

Heptavalent Symmetric Graphs with Certain Conditions

A graph Γ is said to be symmetric if its automorphism group Aut(Γ)acts transitively on the arc set of Γ.We show that if Γ is a finite connected heptavalent symmetric graph with solvable stabilizer admitting a vertex-transitive non-abelian simple group G of automorphisms,then either G is normal in Aut(Γ),or Aut(Γ)contains a non-abelian simple normal subgroup T such that G≤T and(G,T)is explicitly given as one of 11 possible exceptional pairs of non-abelian simple groups.If G is arc-transitive,then G is always normal in Aut(r),and if G is regular on the vertices of Γ,then the number of possible exceptional pairs(G,T)is reduced to 5.     

涉及微分多项式的正规定则（英文）

本文获得如下结果：设φ（z)为区域G内一不恒为零的亚纯函数，a1(z),a2(z),．…，ak(z)为区域G内的全纯函数，F＝{f}为G内一亚纯函数族，若对每一f∈F，在G内恒有f(z)，f(z)≠0,f^(k)(z) a1(z)f^(f-1)(z) … ak(z)f(z)≠φ（z),且与φ（z)没有公共极点，则F在G内正规。     

二面体群的小度数Cayley图的同构类的计数

设G是有限群,S是G的一个不包含单位元的非空子集且满足S^-1=S,定义群G关于S一个的Cayley图x=Cay(G,S)如下：V(X)=G,E(X)=｛(g,sg)｜g∈G,s∈S}.对于素数P,本文给出了2p阶的二面体群的3度和4度Cayley图的同构类的个数.     

THE EXISTENCE OF RADIAL LIMITS OF ANALYTIC FUNCTIONS IN BANACH SPACES

Let X be a comPlex Banach space and let D be the open unit disc in the complex plane.We shall denote by H"(D, X) the Banach space consisting of all uniformly bounded X-vaued analytic functions defined on D equipped with the norm llflloo = suP lIf(z)Il. Az eDcomplex Banach space X is said to have the analytic Radon-NikOdym property if eachelemellt f E Hoo(D,X) has radial limits almost everywhere on the torus T = {e": 0 E[0, 2x]} (see [1]), this means that for almost all 0 C [0,27l, 9W…     

Invariant random subgroups of linear groups

Let Γ < GL _n (_F) be a countable non-amenable linear group with a simple, center free Zariski closure. Let Sub(Γ) denote the space of all subgroups of Γ with the compact, metric, Chabauty topology. An invariant random subgroup (IRS) of Γ is a conjugation invariant Borel probability measure on Sub(Γ). An IRS is called non-trivial if it does not have an atom in the trivial group, i.e. if it is non-trivial almost surely. We denote by IRS⁰(Γ) the collection of all non-trivial IRS on Γ.

Theorem 0.1: With the above notation, there exists a free subgroup F < Γ and a non-discrete group topology on Γ such that for every μ ∈ IRS⁰(Γ) the following properties hold:

μ-almost every subgroup of Γ is open

• F ·Δ = Γ for μ-almost every Δ ∈ Sub(Γ).
• F ∩ Δ is infinitely generated, for every open subgroup. In particular, this holds for μ-almost every Δ ∈ Sub(Γ).
• The map

Φ: (Sub(Γ), μ) → (Sub(F),Φ*μ) Δ → Δ ∩ F is an F-invariant isomorphism of probability spaces.A more technical version of this theorem is valid for general countable linear groups. We say that an action of Γ on a probability space, by measure preserving transformations, is almost surely non-free (ASNF) if almost all point stabilizers are non-trivial.Corollary 0.2: Let Γ be as in the Theorem above. Then the product of finitely many ASNF Γ-spaces, with the diagonal Γ action, is ASNF.Corollary 0.3: Let Γ < GL_n(F) be a countable linear group, A Δ Γ the maximal normal amenable subgroup of Γ — its amenable radical. If μ ∈ IRS(Γ) is supported on amenable subgroups of Γ, then in fact it is supported on Sub(A). In particular, if A(Γ) = <e> then Δ = <e>, μ almost surely.