A new characterization of L 2(r) by their Sylow numbers

Let G be a finite centerless group, let π(G) be the set of prime divisors of the order of G, and let np(G) be the number of Sylow p-subgroups of G, that is, n_p(G) = |Sylp(G)|. Set NS(G) := {n_p(G)| p ∈π(G)}. In this paper, we are investigating whether L_2(r) is determined up to isomorphism by NS(L_2(r)) when r is prime.     

Bipartition of graph under degree constraints

Let G be a graph, let s be a positive integer, and let X be a subset of V(G). Denote δ(X) to be the minimum degree of the subgraph G[X] induced by X. A partition(X, Y) of V(G) is called s-good if min{δ(X), δ(Y)} s. In this paper, we strengthen a result of Maurer and a result of Arkin and Hassin, and prove that for any positive integer k with 2 k |V(G)|- 2, every connected graph G with δ(G) 2 admits a1-good partition(X, Y) such that |X| = k and |Y| = |V(G)|- k, and δ(X) + δ(Y) δ(G)- 1.     

Solvable base change and Rankin-Selberg convolutions

Let π and π′ be unitary automorphic cuspidal representations of GL_n(A_E) and GL_m(A_F), and let E and F be solvable Galois extensions of Q of degrees ? and ?′, respectively. Using the fact that the automorphic induction and base change maps exist for E and F, and assuming an invariance condition under the actions of the Galois groups, we attach to the pair(π, π′) a Rankin-Selberg L-function L(s, π×E,Fπ′) for which we prove a prime number theorem. This gives a method for comparing two representations that could be defined over completely different extensions, and the main results give a measure of how many cuspidal components the two representations π and π′ have in common when automorphically induced down to the rational numbers. The proof uses the structure of the Galois group of the composite extension EF and the character groups attached to the fields via class field theory. The second main theorem also gives an indication of when the base change of π up to the composite extension EF remains cuspidal.     

The extension of the first main theorem for π-blocks

Let π be a set of primes and G a π-separable group. Isaacs defines the Bπ characters, which can be viewed as the "π-modular" characters in G, such that the Bp' characters form a set of canonical lifts for the p-modular characters. By using Isaacs' work, Slattery has developed some Brauer's ideals of p-blocks to the π-blocks of a finite π-separable group, generalizing Brauer's three main theorems to the π-blocks. In this paper, depending on Isaacs' and Slattery's work, we will extend the first main theorem for π-blocks.     

NEIGHBORHOOD UNION OF INDEPENDENT SETS AND HAMILTONICITY OF CLAW-FREE GRAPHS

Let G be a graph,for any u∈V(G),let N(u) denote the neighborhood of u and d(u)=|N(u)| be the degree of u. For any U V(G) ,let N(U)=Uu,∈UN(u), and d(U)=|N(U)|.A graph G is called claw-free if it has no induced subgraph isomorphic to K1.3. One of the fundamental results concerning cycles in claw-free graphs is due to Tian Feng,et al. : Let G be a 2-connected claw-free graph of order n,and d(u) d(v) d(w)≥n-2 for every independent vertex set {u,v,w} of G, then G is Hamiltonian. It is proved that, for any three positive integers s ,t and w,such that if G is a (s t w-1)connected claw-free graph of order n,and d(S) d(T) d(W)＞n-(s t w) for every three disjoint independent vertex sets S,T,W with |S |=s, |T|=t, |W|=w,and S∪T∪W is also independent ,then G is Hamiltonian. Other related results are obtained too.     

On a Conjecture of S.Halperin

If X is a finite simply connected CW complex, then H_*(X,Q) is finitedimensional, let n_X = max{i|H_i(X,Q)≠0}. On the other hand,π_i(X) is the directsum of finitely many copies of Z and finite Abelian group. We call an interval[k,l] a torsion gap for X if π_k(X) and π_l (X) both coutain copies of Z, andπ_i (X)(k相似文献   

A Note on Matchings of Bipartite Graphs

In this paper it is proved that let G(X,Y) be a bipartite graph with A(?)X and B(?)Y, and let k be an integer such that max{|A|, |B|}≤k≤min {|X|,|Y|}, then G contains a matching of k edges saturating both A and B iff for all S(?)X, (i) min{|N(S)| |X|-k, |N_B(S)| |X|-|B|}≥|S| and (ii) min{|N(S∩A)|,|N_B(S∩A)| k-|B|}≥|B∩A|, where N_B(S∩A)=N(S∩A∩B).It extends the theorems of Ore, Chen, Hoffman and Kuhn.     

Finite groups with seminormal Sylow subgroups

In this paper, we prove the following theorem： Let p be a prime number, P a Sylow psubgroup of a group G and π = π（G） ／ {p}. If P is seminormal in G, then the following statements hold： 1） G is a p-soluble group and P＇ ≤ Op（G）; 2） lp（G） ≤ 2 and lπ（G） ≤ 2; 3） if a π-Hall subgroup of G is q-supersoluble for some q ∈ π, then G is q-supersoluble.     

On σ-semipermutable Subgroups of Finite Groups

Let σ = {σ_i|i ∈ I } be some partition of the set of all primes P, G a finite group andσ(G) = {σ_i |σ_i ∩π(G) = ?}. A set H of subgroups of G is said to be a complete Hall σ-set of G if every member = 1 of H is a Hall σ_i-subgroup of G for some σ_i ∈σ and H contains exactly one Hallσ_i-subgroup of G for every σ_i ∈σ(G). A subgroup H of G is said to be: σ-semipermutable in G with respect to H if H H_i~x= H_i~xH for all x ∈ G and all H_i ∈ H such that(|H|, |H_i|) = 1; σ-semipermutable in G if H is σ-semipermutable in G with respect to some complete Hall σ-set of G. We study the structure of G being based on the assumption that some subgroups of G are σ-semipermutable in G.     

Characterization of L2(16) by Τe(L2(16))

Let G be a group and πe(G) the set of element orders of G.Let k∈πe(G) and m k be the number of elements of order k in G.Letτe(G)={mk|k∈πe(G)}.In this paper,we prove that L2(16) is recognizable byτe (L2(16)).In other words,we prove that if G is a group such that τe(G)=τe(L2(16))={1,255,272,544,1088,1920},then G is isomorphic to L2(16).     

用$\tau_e(L_2(16))$刻画$L_2(16)$

Let G be a group and πe(G) the set of element orders of G.Let k∈πe(G) and m k be the number of elements of order k in G.Letτe(G)={mk|k∈πe(G)}.In this paper,we prove that L2(16) is recognizable byτe (L2(16)).In other words,we prove that if G is a group such that τe(G)=τe(L2(16))={1,255,272,544,1088,1920},then G is isomorphic to L2(16).     

Abelian quotients and orbit sizes of linear groups

Let G be a finite group,and let V be a completely reducible faithful finite G-module(i.e.,G ≤GL(V),where V is a finite vector space which is a direct sum of irreducible G-submodules).It has been known for a long time that if G is abelian,then G has a regular orbit on V.In this paper we show that G has an orbit of size at least |G/G′| on V.This generalizes earlier work of the authors,where the same bound was proved under the additional hypothesis that G is solvable.For completely reducible modules it also strengthens the 1989 result |G/G′| |V| by Aschbacher and Guralnick.     

A RECOGNITION OF SIMPLE GROUPS PSL（3, q） BY THEIR ELEMENT ORDERS

For any group G, denote byπe(G) the set of orders of elements in G. Given a finite group G, let h(πe (G)) be the number of isomorphism classes of finite groups with the same set πe(G) of element orders. A group G is called k-recognizable if h(πe(G)) = k <∞, otherwise G is called non-recognizable. Also a 1-recognizable group is called a recognizable (or characterizable) group. In this paper the authors show that the simple groups PSL(3,q), where 3 < q≡±2 (mod 5) and (6, (q-1)/2) = 1, are recognizable.     

Finite groups whose n-maximal subgroups are σ-subnormal

Let σ = {σ_i | i ∈ I} be some partition of the set of all primes P. A set H of subgroups of G is said to be a complete Hall σ-set of G if every member ≠ 1 of H is a Hall σ_i-subgroup of G, for some i ∈ I, and H contains exactly one Hall σ_i-subgroup of G for every σ_i ∈σ(G). A subgroup H of G is said to be: σ-permutable or σ-quasinormal in G if G possesses a complete Hall σ-set H such that HA~x= A~xH for all A ∈ H and x ∈ G:σ-subnormal in G if there is a subgroup chain A = A_0≤A_1≤···≤ A_t = G such that either A_(i-1)■A_i or A_i/(A_(i-1))A_i is a finite σ_i-group for some σ_i ∈σ for all i = 1,..., t.If M_n M_(n-1) ··· M_1 M_0 = G, where Mi is a maximal subgroup of M_(i-1), i = 1, 2,..., n, then M_n is said to be an n-maximal subgroup of G. If each n-maximal subgroup of G is σ-subnormal(σ-quasinormal,respectively) in G but, in the case n 1, some(n-1)-maximal subgroup is not σ-subnormal(not σ-quasinormal,respectively) in G, we write m_σ(G) = n(m_(σq)(G) = n, respectively).In this paper, we show that the parameters m_σ(G) and m_(σq)(G) make possible to bound the σ-nilpotent length l_σ(G)(see below the definitions of the terms employed), the rank r(G) and the number |π(G)| of all distinct primes dividing the order |G| of a finite soluble group G. We also give the conditions under which a finite group is σ-soluble or σ-nilpotent, and describe the structure of a finite soluble group G in the case when m_σ(G) = |π(G)|. Some known results are generalized.     

ON INNERπ'-CLOSED GROUPS AND NORMAL π-COMPLEMENTS

In this paper, the author classifies the finite inner π′-closed groups, and proves the following results1. If each proper subgroup K of a group G is weak π-homogeneous and weak π′-homogeneous, then G is a Schmidt group, or a direct product of two Hall subgroups.2. If G is a weak π-homogeneous group, then G is π′-closed if one of the following statements is true: (1)Each π-subgroup of G is 2-closed. (2) Each π-subgroup of G is 2′-closed.3. Let G be a group and π be a set of odd primes. If N_G(Z(J(P))) has a normal π-completement for a Sytow p-subgroup of G with prime ρ in π then so does G.     

Properties of π-skew Graphs with Applications

The skewness of a graph G, denoted by sk(G), is the minimum number of edges in G whose removal results in a planar graph.It is an important parameter that measures how close a graph is to planarity, and it is complementary, and computationally equivalent, to the Maximum Planar Subgraph Problem.For any connected graph G on p vertices and q edges with girth g, one can easily verify that sk(G) ≥π(G), where π(G) =[q-g/(g-2)(p-2)], and the graph G is said to be π-skew if equality holds.The concept of π-skew was first proposed by G.L.Chia and C.L.Lee.The π-skew graphs with girth 3 are precisely the graphs that contain a triangulation as a spanning subgraph.The purpose of this paper is to explore the properties of π-skew graphs.Some families of π-skew graphs are obtained by applying these properties, including join of two graphs, complete multipartite graphs and Cartesian product of two graphs.We also discuss the threshold for the existence of a spanning triangulation.Among other results some sufficient conditions regarding the regularity and size of a graph, which ensure a spanning triangulation, are given.     

由双边群相关Hopf模诱导的Morita等价

Let H be a Hopf π-coalgebra over a commutative ring k with bijective antipode S, and A and B right π-H-comodulelike algebras. We show that the pair of adjoint functors （F3 = A  Bop A□ HBop -,G3 = （-）coH） between the categories A□HBopM and AMπB-H is a pair of inverse equivalences, when A is a faithfully flat π-H-Galois extension. Furthermore, the categories Moritaπ-H（A,B） and Morita □π-H（AcoH,BcoH） are equivalent, if A and B are faithfully flat π-H-Galois extensions.     

More about the kernel convergence and the ideal convergence

Let I 2N be an ideal and let XI = span{χI ： I ∈ I}, and let pI be the quotient norm of l∞/XI. In this paper, we show first that for each proper ideal I 2N, the ideal convergence deduced by I is equivalent to pI-kernel convergence. In addition, let K = {x＊oχ（·） ： x＊∈ p（e）}, where p（x） = lim supn→∞1/n（∑k=1n｜x（k）｜, and let Iμ = {A N ： μ（A） = 0} for all μ = x＊oχ（·） ∈ K. Then Iμ is a proper ideal. We also show that the ideal convergence deduced by the proper ideal Iμ, the p-kernel convergence and the statistical convergence are also equivalent.     

On the adjacent-vertex-strongly-distinguishing total coloring of graphs

For any vertex u∈V(G), let T_N(U)={u}∪{uv|uv∈E(G), v∈v(G)}∪{v∈v(G)|uv∈E(G)}and let f be a total k-coloring of G. The total-color neighbor of a vertex u of G is the color set C_f(u)={f(x)|x∈TN(U)}. For any two adjacent vertices x and y of V(G)such that C_f(x)≠C_f(y), we refer to f as a k-avsdt-coloring of G("avsdt"is the abbreviation of"adjacent-vertex-strongly- distinguishing total"). The avsdt-coloring number of G, denoted by X_(ast)(G), is the minimal number of colors required for a avsdt-coloring of G. In this paper, the avsdt-coloring numbers on some familiar graphs are studied, such as paths, cycles, complete graphs, complete bipartite graphs and so on. We proveΔ(G) 1≤X_(ast)(G)≤Δ(G) 2 for any tree or unique cycle graph G.     

The Maximum Trigonometric Degrees of Quadrature Formulae with Prescrib ed No des

The purpose of this paper is to study the maximum trigonometric degree of the quadrature formula associated with m prescribed nodes and n unknown additional nodes in the interval（-π, π]. We show that for a fixed n, the quadrature formulae with m and m ＋ 1 prescribed nodes share the same maximum degree if m is odd. We also give necessary and sufficient conditions for all the additional nodes to be real, pairwise distinct and in the interval（-π, π] for even m, which can be obtained constructively. Some numerical examples are given by choosing the prescribed nodes to be the zeros of Chebyshev polynomials of the second kind or randomly for m ≥ 3.