关于有限群的S-半置换子群

Let d be the smallest generator number of a finite p-group P and let Md（P） = {P1,...,Pd} be a set of maximal subgroups of P such that ∩di=1 Pi = Φ（P）. In this paper, we study the structure of a finite group G under the assumption that every member in Md（Gp） is S-semipermutable in G for each prime divisor p of ｜G｜ and a Sylow p-subgroup Gp of G.     

子群的S-条件置换性对有限群结构的影响

Let G be a finite group. Fix a prime divisor p of IGI and a Sylow p-subgroup P of G, let d be the smallest generator number of P and Ma（P） denote a family of maximal subgroups P1, P2 , Pd of P satisfying ∩^di=1 Pi = Ф（P）, the Frattini subgroup of P. In this paper, we shall investigate the influence of s-conditional permutability of the members of some fixed .Md（P） on the structure of finite groups. Some new results are obtained and some known results are generalized.     

具有弱$s$-置换嵌入子群的有限群

Let P be a Sylow p-subgroup of a group G with the smallest generator number d,where p is a prime.Denote by M_d(P) = {P_1,P_(2,...,)P_d} a set of maximal subgroups of P such that φ(P) = ∩_(n=1)~dP_n.In this paper,we investigate the structure of a finite group G under the assumption that the maximal subgroups in M_d(P) are weakly s-permutably embedded in G,some interesting results are obtained which generalize some recent results.Finally,we give some further results in terms of weakly s-permutably embedded subgroups.     

L(d1, d2,..., dt)-Number λ(Cn; d1, d2,...,dt) of Cycles

An L(d1,d2,...,dt)-labeling of a graph G is a function f from its vertex set V(G) to the set {0, 1,..., k} for some positive integer k such that {f(x) - f(y)| ≥ di, if the distance between vertices x and y in G is equal to i for i = 1,2,...,t. The L(d1,d2,...,dt)-number λ(G;d1,d2,... ,dt) of G is the smallest integer number k such that G has an L(d1,d2,... ,dt)labeling with max{f(x)|x ∈ V(G)} = k. In this paper, we obtain the exact values for λ(Cn; 2, 2,1) and λ(Cn; 3, 2, 1), and present lower and upper bounds for λ(Cn; 2,..., 2,1,..., 1)     

On p-nilpotency and minimal subgroups of finite groups

We call a subgroup H of a finite group G c-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K ≤ core(H). In this paper it is proved that a finite group G is p-nilpotent if G is S4-free and every minimal subgroup of P n GN is c-supplemented in NG(P), and when p = 2 P is quaternion-free, where p is the smallest prime number dividing the order of G, P a Sylow p-subgroup of G. As some applications of this result, some known results are generalized.     

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.     

A sufficient condition for planar graphs with maximum degree 8 to be 9-totally colorable

A total k-coloring of a graph G is a coloring of V(G) ∪ E(G) using k colors such that no two adjacent or incident elements receive the same color. The total chromatic number χ'(G) is the smallest integer k such that G has a total k-coloring. It is known that if a planar graph G has maximum degree Δ≥ 9, then χ'(G) = Δ + 1. In this paper, we prove that if G is a planar graph with maximum degree 8 and without a fan of four adjacent 3-cycles, then χ'(G) = 9.     

圈的$L(d_1,d_2,\ldots,d_t)$-数$\lambda(C_n;d_1,d_2,\ldots,d_t)$

An L（d0,d2,...,dt）-labeling of a graph G is a function f from its vertex set V（G） to the set {0,1,..., k} for some positive integer k such that If（x） - f（y）l ≥di, if the distance between vertices x and y in G is equal to i for i = 1,2,...,t. The L（d1,d2,...,dt）-number λ（G;d1,d2,... ,dt） of G is the smallest integer number k such that G has an L（d1,d2,...,dr）- labeling with max{f （x）｜x ∈ V（G）} = k. In this paper, we obtain the exact values for λ（Cn; 2, 2, 1） and λ（Cn; 3, 2, 1）, and present lower and upper bounds for λ（Cn; 2,..., 2, 1,..., 1）     

Neighbor sum distinguishing total colorings of graphs with bounded maximum average degree

A proper [h]-total coloring c of a graph G is a proper total coloring c of G using colors of the set [h]={1,2,...,h}.Letw(u) denote the sum of the color on a vertex u and colors on all the edges incident to u.For each edge uv∈E(G),if w(u)≠w(v),then we say the coloring c distinguishes adjacent vertices by sum and call it a neighbor sum distinguishing [h]-total coloring of G.By tndi(G),we denote the smallest value h in such a coloring of G.In this paper,we obtain that G is a graph with at least two vertices,if mad(G)3,then tndi∑(G)≤k+2 where k=max{Δ(G),5}.It partially con?rms the conjecture proposed by Pil′sniak and Wozniak.     

Neighbor sum distinguishing edge coloring of subcubic graphs

A proper edge-k-coloring of a graph G is a mapping from E(G) to {1, 2,..., k} such that no two adjacent edges receive the same color. A proper edge-k-coloring of G is called neighbor sum distinguishing if for each edge uv ∈ E(G), the sum of colors taken on the edges incident to u is different from the sum of colors taken on the edges incident to v. Let χ_Σ'(G) denote the smallest value k in such a coloring of G. This parameter makes sense for graphs containing no isolated edges(we call such graphs normal). The maximum average degree mad(G) of G is the maximum of the average degrees of its non-empty subgraphs. In this paper, we prove that if G is a normal subcubic graph with mad(G) 5/2,then χ_Σ'(G) ≤ 5. We also prove that if G is a normal subcubic graph with at least two 2-vertices, 6 colors are enough for a neighbor sum distinguishing edge coloring of G, which holds for the list version as well.     

有限群的素数幂阶S-拟正规嵌入子群

设G为有限p-可解群,其中p为|G|的奇素因子.若P为G的Sylow p-子群且最小生成系含d个元素.考虑集合M_d(P)={P_1,…,P_d},其中P_1,…,P_d是P的极大子群且满足(?)P_i=φ(P).证明了若M_d(P)中每个元在G中是S-拟正规嵌入的,则G为p-超可解群.作为应用,还得到了一些进一步的结论.     

$p$-子群具有$p$-超可解正规化子的有限群

在已有研究中,对于$p$-子群的正规化子而言,它的$p$-幂零性质对有限$p$-幂零群的结构具有重要的影响. 本文中, 设$P$是群$G$的西罗$p$-子群, $1\leq p^d<|P|$, 对于$P$的每个阶为$p^d$的正规子群$H$H,将$N_G(H)$的$p$-幂零性质减弱为$p$-超可解性质,结合$H$的弱$M$-可补充性质,探究$p$-超可解群的结构.同时,在$N_G(P)$是$p$-幂零的条件下,利用子群$K$的弱$M$-可补充条件研究群的$p$-幂零性质,其中$K_p\leq K$且$P''\leq K_p\leq \Phi(P)$. $K_p$是$K$的西罗$p$-子群.在一定程度上,主要结果推广了Frobenius定理.     

High-Dimensional Shape Fitting in Linear Time

Let $P$ be a set of $n$ points in $\Re^d$. The {\em radius} of a $k$-dimensional flat ${\cal F}$ with respect to $P$, which we denote by ${\cal RD}({\cal F},P)$, is defined to be $\max_{p \in P} \mathop{\rm dist}({\cal F},p)$, where $\mathop{\rm dist}({\cal F},p)$ denotes the Euclidean distance between $p$ and its projection onto ${\cal F}$. The $k$-flat radius of $P$, which we denote by ${R^{\rm opt}_k}(P)$, is the minimum, over all $k$-dimensional flats ${\cal F}$, of ${\cal RD}({\cal F},P)$. We consider the problem of computing ${R^{\rm opt}_k}(P)$ for a given set of points $P$. We are interested in the high-dimensional case where $d$ is a part of the input and not a constant. This problem is NP-hard even for $k = 1$. We present an algorithm that, given $P$ and a parameter $0 < \eps \leq 1$, returns a $k$-flat ${\cal F}$ such that ${\cal RD}({\cal F},P) \leq (1 + \eps) {R^{\rm opt}_k}(P)$. The algorithm runs in $O(nd C_{\eps,k})$ time, where $C_{\eps,k}$ is a constant that depends only on $\eps$ and $k$. Thus the algorithm runs in time linear in the size of the point set and is a substantial improvement over previous known algorithms, whose running time is of the order of $d n^{O(k/\eps^c)}$, where $c$ is an appropriate constant.     

2-极大子群的$F_s$-拟正规性

$F$是一个群系. $G$的子群$H$在$G$中称为$F_s$-拟正规的,如果存在$G$的正规子群$T$,使得$HT$在$G$中是$s$-置换的并且$(H\cap T)H_G/H_G$包含在$G/H_G$的$F$超中心$Z^F_\infty(G/H_G)$中.本文利用$F_s$-拟正规子群研究了有限群的结构.获得了某些新的结果.     

p-超循环嵌入子群的一个判别准则

令E是有限群G的一个正规子群,且U是所有有限超可解群的集合.E称为在G中是p-超循环嵌入的,如果E的每个pd-阶的G-主因子是循环的.G的子群H称为在G中是U-Φ-可补充的,如果存在G的一个次正规子群T,使得G=HT,且(H∩T)H_G/H_G≤Φ/(H/H_G)Z_U(G/H_G),其中Z_U(G/H_G)是商群G/H_G的U-超中心.作者证明,如果E的一些p-子群在G中是U-Φ-可补充的,那么E在G中是p-超循环嵌入的.作为应用,得到了有限群是p-超可解的若干判断准则,并且推广了一些已知的结果.     

Wreath Product of Set-Valued Functors and Tensor Multiplication

Considering the wreath product functor $G wr H:{\cal A} wr^G{\cal B} \rightarrow \SET$ of functors $G: {\cal A}\rightarrow \SET$ and $H: {\cal B}\rightarrow \SET$ over small categories $ {\cal A}$ and $ {\cal B}$, we prove that if tensor multiplication by the functor $G\wrr H$ preserves $ {\cal D}$-limits, where ${\cal D}$ is a small category, then tensor multiplication by $G$ preserves ${\cal D}$-limits, and if tensor multiplication by the functor $G wr H$ preserves ${\cal D}$-limits of representables then tensor multiplications by $G$ and $H$ preserve $ {\cal D}$-limits of representables. We also study flatness and pullback flatness of the wreath product of set-valued functors.     

ERGODIC THEOREMS FOR NON-LIPSCHITZIAN SEMIGROUPS WITHOUT CONVEXITY

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