每个子群都c-正规的有限群

在本文中我们研究有限CN-群, 即每个子群都c-正规的有限群. 我们得到以下结果:群G是CN-群当且仅当G^∈的每个子群都在G中正规.群G是CN-群当且仅当G可解且c-正规性是传递的.设p是一个奇素数, P是一个p-群, 则P是一个CN-群当且仅当Φ(P)≤Z(P).我们也得到了一些CN-群的直积为CN-群的判别条件.     

有限CN-群与有限c-可补群*

有限群G的子群H称为G的c-可补子群(c-正规子群),如果存在G的子群(正规子群)N, 使得 G = NH 且 N\cap H \leq H_G,这里 H_G =\bigcap\limits_g\in G H^g 是 H 在 G 中的核.每个子群都c-可补(c-正规)的有限群称为有限c-可补群(CN-群).本文研究有限CN-群与有限c-可补群, 获得了CN- 群与c-可补群的一些新的结果.特别地, 在方法上有一定的创新, 完善近期关于CN-群的研究.     

有限ATI-群的类保持Coleman自同构

设G是一个有限群,对G的任意阿贝尔子群A及任意g∈G,若A∩A~g=1或A,则称G为一个ATI-群.本文证明了,对任意p∈τ(G),如果ATI-群G的一个p-方幂阶类保持自同构在G的任意Sylow子群上的限制等于G的某个内自同构的限制,则它必定是一个内自同构.作为该结果的一个直接推论,我们也证明了有限ATI-群G有正规化性质.     

有限秩的幂零p-群的p-自同构

设G是一个有限秩的幂零p-群,α和β是G的两个p-自同构,记I= ((αβ(g))(βα(g))-1)|g∈G),则(i)当I是有限循环群时,α和β生成一个有限P-群; (ii)当I是拟循环p-群时,α和β生成一个可解的剩余有限P-群,它是有限生成的无挠幂零群被有限p-群的扩张．     

有限秩的幂零群的自同构

设幂零群G=KP=PK,其中P是有限秩的幂零π-群,K是G的有限秩的π′-自由的正规子群.π不属于K的谱Sp(K),设1=ζ0Gζ1G…ζcG=G是G的上中心列,α和β是G的两个自同构,把α和β在每个商因子ζiG/ζ(i—1)G上的诱导自同构分别记为αi和βi,记Ii:=Im(αiβi—βiαi),则(i)当每个Ii都是有限循环群,并且I:=〈(αβ(g))(βα(g))(-1)|g∈G〉是G的有限子群时,α和β生成一个可解的几乎Abel群.(ii)当每个Ii或者是有限循环群,或者是秩1的可除群,或者是C⊕D,其中C是循环群,D是秩1的可除群,或者是无挠的局部循环群,或者Ii有正规子群列1JiIi,其商因子分别为有限循环群,无挠的局部循环群,或者Ii=D⊕Ji,其中D是秩1的可除群,Ji为无挠的局部循环群,或者Ii有正规列1KiJiIi,其商因子分别为有限循环群,秩1的可除群,无挠的局部循环群时,β和β生成一个可解的NAF-群.特别地,如果α和β是A的两个π′-自同构,那么(iii)当每个Ii都是有限循环群,并且I:=〈(αβ(g))(βα(g))(-1)|g∈G〉是有限群时,α和β生成的群是有限幂零π-群被有限Abelπ′-群的扩张.(iv)当每个Ii或者是有限循环群,或者是秩1的可除群,或者是C⊕D,其中C是循环群,D是秩1的可除群时,α和β生成一个可解的剩余有限π∪π′-群,它是有限生成的无挠幂零群被有限可解π∪π′-群的扩张.(v)当每个Ii或者是有限循环群,或者是秩1的可除群,或者是C⊕D,其中C是循环群,D是秩1的可除群,或者是无挠的局部循环群,或者Ii有正规子群列1JiIi,其商因子分别为有限循环群,无挠的局部循环群,或者Ii=D⊕Ji,其中D是秩1的可除群,Ji为无挠的局部循环群,或者Ii有正规列1KiJiIi,其商因子分别为有限循环群,秩1的可除群,无挠的局部循环群时,α和β生成一个可解的剩余有限π∪π′-群,它的幂零长度至多是4.当K是FC-群时,在情形(v)中,α和β生成的群是有限生成的无挠幂零群被有限可解π∪π′-群的扩张.此外,如果G=KP,K是一个FC-群,对G的下中心列考虑了类似的问题,得到了对偶的结果.     

有限群共轭类长的一个问题

设A和B都是有限群G的子群且G=AB.若A是G的次正规子群,且对每个p∈π(G)以及每个素数幂阶的p′-元x∈A∪B,p~2均不整除|x~G|,则G为超可解群.这个结果正面解答了由石向东,韦华全和马儇龙于2013年提出的一个问题,统一推广了由刘晓蕾于2011年得到的三个定理.     

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.     

n-极大子群为共轭可换的有限群

群G的子群H称为G的共轭可换子群,若HHg=HgH,对任意g∈G都成立,本文考查了n-极大子群为共轭可换时对有限群构造的影响.     

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.     

有限群为超可解群的充要条件

用置换条件刻画有限可解群的超可解性已有大量结果,本文的目的是给出另外一些有限可解群为超可解的充要条件。其主要结果是: 1.设G是满足置换条件的有限可解群,则G是超可解群当且仅当如下条件之一成立。 1)G的2-Sylow子群G_2的换位子群G_2′G. 2)G有正规2-补。 2.设G是有限可解群,则G超可解当且仅当G和G′均满足置换条件.     

Strong skew commutativity preserving maps on rings with involution

Let R be a unital *-ring with the unit I.Assume that R contains a symmetric idempotent P which satisfies ARP = 0 implies A = 0 and AR(I-P) = 0 implies A = 0.In this paper,it is shown that a surjective map Φ:R→R is strong skew commutativity preserving(that is,satisfiesΦ(A)Φ(B)-Φ(B)Φ(A)~w= AB-BA~w for all A,B∈R) if and only if there exist a map f:R→Z_s(R)and an element Z∈Z_s(R) with Z~2=I such that Φ(A)=ZA +f(A) for all A∈R,where Z_s(R) is the symmetric center of R.As applications,the strong skew commutativity preserving maps on unital prime *-rings and von Neumann algebras with no central summands of type I_1 are characterized.     

Nonlinear skew lie triple derivations between factors

Let A be a factor.For A,B∈A,define by [A,B]_*=AB-BA~* the skew Lie product of A and B.In this article,it is proved that a map Φ:A→A satisfies Φ([[A,B]_*,C]_*)=[[Φ(A),B]_*,C]_w+[[A,Φ(B)]_*,C]_*+[[A,B]_*,Φ(C)]_* for all A,B,C∈A if and only if Φ is an additive *-derivation.     

Equivalent characterization of centralizers on <Emphasis Type="Italic">B</Emphasis>(<Emphasis Type="Italic">H</Emphasis>)

Let H be a Hilbert space with dim H≥2 and Z∈ß(H) be an arbitrary but fixed operator. In this paper we show that an additive map Φ:ß(H) → ß(H) satisfies Φ(AB)=Φ(A)B=AΦ(B) for any A,B∈ß(H) with AB=Z if and only if Φ(AB)=Φ(A)B=AΦ(B), ∀A, B∈ß(H), that is, Φ is a centralizer. Similar results are obtained for Hilbert space nest algebras. In addition, we show that Φ(A²)=AΦ(A)=Φ(A)A for any A∈ ß(H) with AA²=0 if and only if Φ(A)=AΦ(I)=Φ(I)A, ∀A∈ß(H), and generalize main results in Linear Algebra and its Application, 450, 243-249 (2014) to infinite dimensional case. New equivalent characterization of centralizers on ß(H) is obtained.     

Jacobian-determinant equation in the critical Besov spaces

Let Ω be a bounded domain in R~n with smooth boundary. Here we consider the following Jacobian-determinant equation det u(x)=f(x),x∈Ω;u(x)=x,x∈?Ω where f is a function on Ω with min_Ω f = δ 0 and Ωf(x)dx = |Ω|. We prove that if f ∈B_(p1)~(np)(Ω) for some p∈(n,∞), then there exists a solution u ∈ B_(p1)~(np+1)(Ω)C~1(Ω) to this equation. On the other hand, we give a simple example such that u ∈ C_0~1(R~2, R~2) while detu does not lie in B_(p1)~(2p)(R~2) for any p∞.     

Möbius Homogeneous Hypersurfaces with One Simple Principal Curvature in Sn+1

Let Möb(Sⁿ⁺¹) denote the Möbius transformation group of Sⁿ⁺¹. A hypersurface f:Mⁿ → Sⁿ⁺¹ is called a Möbius homogeneous hypersurface, if there exists a subgroup G ? Möb(Sⁿ⁺¹) such that the orbit G(p)={φ(p)|φ ∈ G}=f(Mⁿ), p ∈ f(Mⁿ). In this paper, we classify the Möbius homogeneous hypersurfaces in Sⁿ⁺¹ with at most one simple principal curvature up to a Möbius transformation.     

Heavy cycles in 2-connected triangle-free weighted graphs

A weighted graph is one in which every edge e is assigned a nonnegative number, called the weight of e. The sum of the weights of the edges incident with a vertex v is called the weighted degree of v, denoted by dw(v). The weight of a cycle is defined as the sum of the weights of its edges. Fujisawa proved that if G is a 2-connected triangle-free weighted graph such that the minimum weighted degree of G is at least d, then G contains a cycle of weight at least 2d. In this paper, we proved that if G is a2-connected triangle-free weighted graph of even size such that dw(u) + dw(v) ≥ 2d holds for any pair of nonadjacent vertices u, v ∈ V(G), then G contains a cycle of weight at least 2d.     

Multiple Positive Solutions for a Nonlinear Elliptic Equation Involving Hardy–Sobolev–Maz'ya Term

In this paper, we study the existence and nonexistence of multiple positive solutions for the following problem involving Hardy–Sobolev–Maz'ya term:-Δu- λu/|y|2=|u|pt-1u/|y|t+ μf(x), x ∈Ω,where Ω is a bounded domain in RN(N ≥ 2), 0 ∈Ω, x =(y, z) ∈ Rk× RN-kand pt =N +2-2t N-2(0 ≤ t ≤2). For f(x) ∈ C1(Ω)\{0}, we show that there exists a constant μ* 0 such that the problem possessesat least two positive solutions if μ∈(0, μ*) and at least one positive solution if μ = μ*. Furthermore,there are no positive solutions if μ∈(μ*, +∞).     

剩余有限minimax可解群的自同构

设G是剩余有限minimax可解群,α是G的自同构且φ:G→G(g→[g,α])是满射,则有以下结果:(1)当α~p=1时,G是幂零类不超过h(p)的幂零群的有限扩张,其中h(p)是只与p有关的函数;(2)当α~4=1时,G存在一个指数有限的特征子群H,使得H″≤Z(H)和C_H(α~2)是Abel群.并且C_G(α~2)和G/[G,α~2]都是Abel群的有限扩张.     

素环上强保持2-Jordan乘积的映射

对于给定的正整数k≥1,环R上的元x,y的k-Jordan乘积定义为{x,y}_k={{x,y}_(k-1),y}_1,其中{x,y}_0=x,{x,y}_1=xy+yx.假设R是包含有单位元与一非平凡幂等元的素环.本文证明了R上的满射f满足{f(x),f(y)}2={x,y}_2对所有x,y∈R成立当且仅当存在λ∈l(R的可扩展中心)且λ~3=1,使得下列之一成立:(1)若R的特征不为2,则f(x)=λx对所有x∈R成立;(2)若R的特征为2,则f(x)=λx+μ(x)对所有x∈R成立,其中μ:R→l是一个映射.作为应用,得到了因子von Neumann代数上保持上述性质映射的结构.