Groups with finiteness conditions on the lower central series of non-normal subgroups

It is known that any locally graded group with finitely many derived subgroups of non-normal subgroups is finite-by-abelian. This result is generalized here, by proving that in a locally graded group G the subgroup \(\gamma _{k}(G)\) is finite if the set \(\{\gamma _{k}(H)\;|\;H\le G,\,H\ntriangleleft G\}\) is finite. Moreover, locally graded groups with finitely many kth terms of lower central series of infinite non-normal subgroups are also completely described.     

On <Emphasis Type="Italic">p</Emphasis>-nilpotence and solubility of groups

Recall a result due to O. J. Schmidt that a finite group whose proper subgroups are nilpotent is soluble. The present note extends this result and shows that if all non-normal maximal subgroups of a finite group are nilpotent, then (i) it is soluble; (ii) it is p-nilpotent for some prime p; (iii) if it is not nilpotent, then the number of prime divisors contained in its order is between 2 and k + 2, where k is the number of normal maximal subgroups which are not nilpotent.     

Lower Bounds on the Number of Cyclic Subgroups in Finite Non-Cyclic Nilpotent Groups

Let $G$ be a finite group and $\mathfrak{c}(G)$ denote the number of cyclic subgroups of $G$. It is known that the minimal value of $\mathfrak{c}$ on the set of groups of order $n$, where $n$ is a positive integer, will occur at the cyclic group $Z_n$. In this paper, for non-cyclic nilpotent groups $G$ of order $n$, the lower bounds of $\mathfrak{c}(G)$ are established.     

Groups with finitely many derived subgroups of non-normal subgroups

A group is called metahamiltonian if all its non-abelian subgroups are normal; it is known that locally soluble metahamiltonian groups have finite derived subgroup. This result is generalized here, by proving that every locally graded group with finitely many derived subgroups of non-normal subgroups has finite derived subgroup. Moreover, locally graded groups having only finitely many derived subgroups of infinite non-normal subgroups are completely described. Received: 25 April 2005     

Groups with few conjugacy classes of non-normal subgroups

The structure of groups with finitely many non-normal subgroups is well known. In this paper, groups are investigated with finitely many conjugacy classes of non-normal subgroups with a given property. In particular, it is proved that a locally soluble group with finitely many non-trivial conjugacy classes of non-abelian subgroups has finite commutator subgroup. This result generalizes a theorem by Romalis and Sesekin on groups in which every non-abelian subgroup is normal.      

Groups whose non-normal subgroups have a transitive normality relation

A group is calledmetahamiltonian if all its non-normal subgroups are abelian. The structure of metahamiltonian groups has been investigated by Romalis and Sesekin. In this paper groups are studied in which every non-normal subgroup has a transitive normality relation.     

Two sufficient conditions for non-normal Cayley graphs and their applications

A Cayley graph Cay(G, S) on a group G is said to be normal if the right regular representation R(G) of G is normal in the full automorphism group of Cay(G, S). In this paper, two sufficient conditions for non-normal Cayley graphs are given and by using the conditions, five infinite families of connected non-normal Cayley graphs are constructed. As an application, all connected non-normal Cayley graphs of valency 5 on A5 are determined, which generalizes a result about the normality of Cayley graphs of valency 3 or 4 on A5 determined by Xu and Xu. Further, we classify all non-CI Cayley graphs of valency 5 on A5, while Xu et al. have proved that As is a 4-CI group.     

Rainb ow Vertex-connection Numb er of Ladder and M¨obius Ladder

A vertex-colored graph G is said to be rainbow vertex-connected if every two vertices of G are connected by a path whose internal vertices have distinct colors, such a path is called a rainbow path. The rainbow vertex-connection number of a connected graph G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G rainbow vertex-connected. If for every pair u, v of distinct vertices, G contains a rainbow u-v geodesic, then G is strong rainbow vertex-connected. The minimum number k for which there exists a k-vertex-coloring of G that results in a strongly rainbow vertex-connected graph is called the strong rainbow vertex-connection number of G, denoted by srvc(G). Observe that rvc(G) ≤ srvc(G) for any nontrivial connected graph G. In this paper, for a Ladder Ln, we determine the exact value of srvc(Ln) for n even. For n odd, upper and lower bounds of srvc(Ln) are obtained. We also give upper and lower bounds of the (strong) rainbow vertex-connection number of M¨obius Ladder.     

广义Mycielski图的邻强边色数和邻点可区别全色数的两个上界

对简单图G,|V(G)|=p,n是自然数,Mn(G)被称为图G的广义Mycielski图,如果V(Mn(G))={v01,v02,…,v0p;v11,v12,…,v1p;…;vn1,vn2,…,vnp},E(Mn(G))=E(G)∪{vijv(i+1)k|v0jv0k∈E(G),1≤j,k≤p,i=0,1,…,n-1}.文中针对简单图G与它的广义Mycielski图之间的关系,给出了G的广义Mycielski图的邻强边色数和邻点可区别全色数的两个上界.     

Algorithmic Modality Analysis for Parabolic Groups

For an algebraic group R acting morphically on an algebraic variety X the modality of the action, mod (R:X), is the maximal number of parameters upon which a family of R-orbits on X depends. Let G be a reductive algebraic group defined over an algebraically closed field K. Let P be a parabolic subgroup of G. Then P acts on its unipotent radical P_u via conjugation and on , the Lie algebra of P_u, via the adjoint action. The modality of P is defined as mod P:=mod (P: ). In this paper we discuss an algorithm which is used to compute upper bounds for mod P along with some results obtained by this algorithm. One is a classification of parabolic subgroups P of simple algebraic groups G of semisimple rank 2 and modality 0. For parabolic subgroups of semisimple rank 3 we present some partial results. This extends the results of Kashin and Popov and Röhrle, where the cases of semisimple rank 0 and 1 are handled. For exceptional groups G we show that P G has modality zero provided the class of nilpotency of P_u is at most two. The analogous result for classical groups is proved by Röhrle. For Borel subgroups B of simple groups we are able to determine the value for mod B in some small rank cases by combining lower bounds for mod B of Röhrle with upper bounds provided by the algorithm.     

广义de Bruijn和Kautz有向图的双向控制集

设G=(V,A)是一个有向图,其中V和A分别表示有向图G的点集和弧集.对集合TV(G),如果对于任意点v∈V(G)\T,都存在点u,w∈T(u,w可能是同一点)使得(u,v),(v,w)∈A(G),则称T是G的一个双向控制集.有向图G的双向控制数γ~*(G)是G的最小双向控制集所含点的数目.提出了广义de Bruijn和Kautz有向图的双向控制数的新上界,改进了以前文献中提出的相关结论.此外,对某些特殊的广义de Bruijn和Kautz有向图,通过构造其双向控制集,进一步改进了它们双向控制数的上、下界.     

Semisimple Types in GLn

This paper is concerned with the smooth representation theory of the general linear group G=GL(F) of a non-Archimedean local field F. The point is the (explicit) construction of a special series of irreducible representations of compact open subgroups, called semisimple types, and the computation of their Hecke algebras. A given semisimple type determines a Bernstein component of the category of smooth representations of G; that component is then the module category for a tensor product of affine Hecke algebras; every component arises this way. Moreover, all Jacquet functors and parabolic induction functors connecting G with its Levi subgroups are described in terms of standard maps between affine Hecke algebras. These properties of semisimple types depend on their special intertwining properties which in turn imply strong bounds on the support of coefficient functions.     

On the number of maximal subgroups of a finite solvable group

For a finite solvable group G and prime number p, we use elementary methods to obtain an upper bound for \mathfrak m_p(G){\mathfrak {m}_{p}(G)} , defined as the number of maximal subgroups of G whose index in G is a power of p. From this we derive an upper bound on the total number of maximal subgroups of a finite solvable group in terms of its order. This bound improves existing bounds, and we identify conditions on the order of a finite solvable group under which this bound is best possible.     

弦图的L(3,2,1)}-标号

图 G 的一个 L(3,2,1)- 标号是指从 V(G) 到非负整数集的一个映射 f, 满足: 当 d_G(u,v)=1 时, |f(u)-f(v)|\geq 3; 当 d_G(u,v)=2 时, |f(u)-f(v)|\geq 2; 当 d_G(u,v)=1 时, |f(u)-f(v)|\geq 1. L(3,2,1)-标号问题就是确定出最小的整数 \lambda_3(G) 使得 G存在最大标号不超过该数的 L(3,2,1)- 标号. 本文研究了弦图的 L(3,2,1)- 标号问题,获得了弦图及其一些子类, 如扇, r- 路,r- 树等的 \lambda_3 数的界.     

关于线图和全图的原子键连通性指数

连通图G的原子键连通性(ABC)指数定义为: ABC(G)=\sum\limits_{uv\in E(G)} \sqrt{\frac{d(u)+d(v)-2}{d(u)d(v)}} , 其中E(G)为图G的边集, d(u) 和d(v)为顶点u和v的度数. 原子键连通性指数是化学图论中比较重要的连通度指数, 最近的研究表明它可以用来研究烷烃的能量信息. 给出了线图和全图的ABC指数的上界和下界, 并且证明了这些界是可达的.     

图的符号星k控制数

引入了图的符号星k控制的概念.设G=（V,E）是一个图,一个函数f：E→{-1,＋1},如果∑e∈E[v]f（e）≥1对于至少k个顶点v∈V（G）成立,则称f为图G的一个符号星k控制函数,其中E（v）表示G中与v点相关联的边集.图G的符号星k控制数定义为γkss（G）=min{∑e∈Ef（e）｜f为图G的符号星k控制函数}.在本文中,我们主要给出了一般图的符号星k控制数的若干下界,推广了关于符号星控制的一个结果,并确定路和圈的符号星k控制数.     

On Solvability of Finite Groups with Few Non-Normal Subgroups

For a finite group G, let v(G) denote the number of conjugacy classes of non-normal subgroups of G and vc(G) denote the number of conjugacy classes of non-normal noncyclic subgroups of G. In this paper, we show that every finite group G satisfying v(G) ≤2|π(G)| or vc(G) ≤ |π(G)| is solvable, and for a finite nonsolvable group G, v(G) = 2|π(G)| +1 if and only if G ? A ₅.     

关于几类图的L(2,1)标号问题

图G的L( 2 ,1 )标号是一个从顶点集V(G)到非负整数集的函数f(x) ,使得若d(x ,y) =1 ,则｜f(x) -f(y) ｜≥ 2 ;若d(x ,y) =2 ,则｜f(x) -f(y) ｜≥ 1 .图G的L( 2 ,1 ) 标号数λ(G)是使得G有max{f(v) ∶v∈V(G) }=k的L( 2 ,1 )标号中的最小数k .Griggs和Yeh猜想对最大度为Δ的一般图G ,有λ(G) ≤Δ2 .本文给出了Kneser图 ,Mycieklski图 ,Descartes图 ,Halin图的λ值的上界 ,并证明了上述猜想对以上几类图成立     

Maximal Subgroups in the Classical Groups Normalizing Solvable Subgroups

Let G be a classical group over an arbitrary field F,acting on an n-dimensional F-space V=V(n,F).All those maximal subgroups of G are classified each of which normalizes a solvable subgroup N of GL(V/F)not lying in F*1v.     

On Jørgensen numbers and their analogs for groups of figure-eight orbifolds

The Jørgensen, Gehring-Martin-Tan, and Tan numbers are defined for every two-generated subgroup of the group PSL(2,C). These numbers arise in necessary discreteness conditions for two-generated subgroups. The Jørgensen number equals 1 for the figure-eight knot group. We calculate the above numbers or give some two-sided bounds of them for this group and groups of hyperbolic orbifolds with singularities along the figure-eight knot.