Average Probability of an Element Being a Generator in the Cyclic Group

All elements in the cyclic group are generated by a generator g. The number of generators of of , namely is known to be Euler’s totient function;however, the average probability of an element being a generator has not been discussed before. Several analytic properties of have been investigated for a long time. However, it seems that some issues still remain unresolved. In this study, we derive the average probability of an element being a generator using previous classical studies.

Vacuum Solutions of Classical Gravity on Cyclic Groups from Noncommutative Geometry

Based on the observation that the moduli of a link variable on a cyclic group modify Connes' distance on this group, we construct several action functionals for this link variable within the framework of noncommutative geometry. After solving the equations of motion, we find that one type of action gives nontrivial vacuum solution for gravity on this cyclic group in a broad range of coupling constants and that such a solution can be expressed with Chebyshev's polynomials.

Locally Transitive Graphs Admitting a Group with Cyclic Sylow Subgroups

All graphs are finite simple undirected and of no isolated vertices in this paper.Using the theory of coset graphs and permutation groups,it is completed that a classification of locally transitive graphs admitting a non-Abelian group with cyclic Sylow subgroups.They are either the union of the family of arc-transitive graphs,or the union of the family of bipartite edge-transitive graphs.

Ideal homomorphic secret sharing schemes over cyclic groups

For a cyclic group G and an access structure A, the sufficient and necessary condition under which A is G ideal homomorphic is given by using the fine representation of the corresponding matroid over the ring Z m . Furthermore, the classification of G ideal homomorphic graphic access structures is shown.

Modular Vector Invariants of Cyclic Groups Z_2

Let G be the finite cyclic group Z_2 and V be a vector space of dimension 2n with basis x_1,...,x_n,y_1,...,y_n over the field F with characteristic 2.If σ denotes a generator of G,we may assume that σ(x_i)= ayi,σ(y_i)= a～-1x_i,where a ∈ F.In this paper,we describe the explicit generator of the ring of modular vector invariants of F[V]～G.We prove that F[V]～G = F[l_i = x_i + ay_i,q_i = x_iy_i,1 ≤ i ≤ n,M_I = X_I + a～-I-Y_I],where I∈An = {1,2,...,n},2 ≤-I-≤ n.

The F-Decomposition of Artinian Modules over Hyper-(Cyclic or Finite)Groups

Let (?) be a formation locally defined by f(p), G ∈(?) and A a ZG-module, where p ∈ $ = { all primes and symbol 88 }. Then a p-main-factor U/V of G is said to be (?)-central in G if G/CG(U/V) ∈f(p). In this paper, we have proved that: let (?) be a locally defined formation consisting of locally soluble groups, G a hyper-(cyclic or finite) locally soluble group and A an artinian ZG-module with all irreducible ZG-factors of A being finite; if G ∈ (?), f(∞) ∈ f(p) , f(p) ≠φ for each p ∈ π, A has an (?)-decomposition.

Splitting Extensions of Abelian by Hyper-(cyclic or finite) Groups (Ⅱ)

Let Fbe a locally defined formation consisting of locally solvable groups, G a hyper-(cyclic or finite) locally solvable group and A a noetherian ZG-module with all irreducible ZG-factors being finite. The following conclusion is obtained: if G∈F, f(∞)f(p), f(p)≠ for each p∈π, and A has no nonzero F-central ZG- images, then any extension E of A by G splits conjugately over A, and A has no nonzero F-central ZG-factors.

FINITE p-GROUPS WHICH CONTAIN A SELF-CENTRALIZING CYCLIC NORMAL SUBGROUP

For any prime p,all finite noncyclic p-groups which contain a self-centralizing cyclic normal subgroup are determined by using cohomological techniques.Some applications are given,including a character theoretic description for such groups.

F-Decomposition of Noetherian Modules over Hyper-(Cyclic or Finite) Groups

Let Fbe a locally defined formation consisting of locally soluble groups, G a hyper-(cyclic or finite) locally soluble group and A a noetherian ZG-module with all irreducible ZG-factors being finite, G∈F, f(∞)f(p), f(p)≠ for each p∈π. The following conclutions are obtained: (1) if there exists a maximal submodule B of A such that A/B is F-central in G and B has no nonzero F-central ZG-factors, then A has an F-decomposition; (2) if there exists an irreducible F-central submodule B of A such that all ZG-composition factors of A/B are F-ecentric, then A has an F-decomposition.

Behavior of Pile Group with Elevated Cap Subjected to Cyclic Lateral Loads

The pile group with elevated cap is widely used as foundation of offshore structures such as turbines, power transmission towers and bridge piers, and understanding its behavior under cyclic lateral loads induced by waves, tide water and winds, is of great importance to designing. A large-scale model test on 3×3 pile group with elevated cap subjected to cyclic lateral loads was performed in saturated silts. The preparation and implementation of the test is presented. Steel pipes with the outer diameter of 114 mm, thickness of 4.5 mm, and length of 6 m were employed as model piles. The pile group was cyclic loaded in a multi-stage sequence with the lateral displacement controlled. In addition, a single pile test was also conducted at the same site for comparison. The displacement of the pile cap, the internal forces of individual piles, and the horizontal stiffness of the pile group are presented and discussed in detail. The results indicate that the lateral cyclic loads have a greater impact on pile group than that on a single pile, and give rise to the significant plastic strain in the soil around piles. The lateral loads carried by each row of piles within the group would be redistributed with loading cycles. The lateral stiffness of the pile group decreases gradually with cycles and broadly presents three different degradation patterns in the test. Significant axial forces were measured out in some piles within the group, owing to the strong restraint provided by the cap, and finally lead to a large settlement of the pile group. These findings can be referred for foundation designing of offshore structures.

Toric Heaps, Cyclic Reducibility, and Conjugacy in Coxeter Groups

In 1986, G.X. Viennot introduced the theory of heaps of pieces as a visualization of Cartier and Foata’s “partially commutative monoids”. These are essentially labeled posets satisfying a few additional properties, and one natural setting where they arise is as models of reduced words in Coxeter groups. In this paper, we introduce a cyclic version of a heap, which loosely speaking, can be thought of as taking a heap and wrapping it into a cylinder. We call this object a toric heap, because we formalize it as a labeled toric poset, which is a cyclic version of an ordinary poset. Defining the category of toric heaps leads to the notion of certain morphisms such as toric extensions. We study toric heaps in Coxeter theory, because a cyclic shift of a reduced word is simply a conjugate by an initial or terminal generator. As such, we formalize and study a framework that we call cyclic reducibility in Coxeter theory, which is closely related to conjugacy. We introduce what it means for elements to be torically reduced, which is a stronger condition than simply being cyclically reduced. Along the way, we encounter a new class of elements that we call torically fully commutative (TFC), which are those that have a unique cyclic commutativity class, and comprise a strictly bigger class than the cyclically fully commutative (CFC) elements. We prove several cyclic analogues of results on fully commutative (FC) elements due to Stembridge. We conclude with how this framework fits into recent work in Coxeter groups, and we correct a minor flaw in a few recently published theorems.

半群CSP_(n,k)的秩和k方幂等元秩

设自然数n≥3,P_(n)和S_(n)是有限链X_(n)上的部分变换半群和对称群.对任意的正整数k满足3≤k≤n,令C_(k)=g_(k)是X_(n)上的k-局部循环群且CSP_(n,k)=C_(k)∪(P_(n)\S_(n)),易证CSP_(n,k),是部分变换半群P_(n)的子半群.通过分析半群CSP_(n,k),的格林关系和幂等元,获得了半群CSP_(n,k),的极小生成集和k方幂等元极小生成集,进一步确定了半群CSP_(n,k),的秩和k方幂等元秩.

有限循环群上奇素数次方根态射的个数

设G为m阶有限循环群,m>1,φ:G→G为群同态。若φ^(k)=1_(G),1_(G)为恒等态射,k为正整数,称φ为G的k次方根态射,求出当k为奇素数时G的k次方根态射的个数。

循环码的一种构造方法研究

运用循环码生成多项式的性质与定义,对循环码新的构造方法进行研究,确定了构造出的循环码的重量分布、重量谱、最小距离和自同构群。新的构造方法在一定条件下比现有文献中的构造方法更易得出循环码的自同构群等性质。

关于循环子群数目超过一半的几乎单群

假设G为有限群,记c(G)为群G的循环子群个数,α(G)为群G的循环子群个数与群阶数的比值,即a(G):=c(G)/|G|。本次研究证明了若G为几乎单群,即S≤G≤Aut(S),这里S为有限非交换单群,则α(G)≥1/2当且仅当G■@A_(5),S_(5),S_(6)。

Quantum Group Symmetry in Hofstadter Problem

We show that there is a quantum Sl_q(2) group symmetry in Hofstadter problem on square lattice.The cyclic representation of the quantum group is discussed and its application for computing the degeneracy density of the model is shown.

On Some Embedment of Groups into Wreath Products

In this paper, we showed how groups are embedded into wreath products, we gave a simpler proof of the theorem by Audu (1991) (see [1]), also proved that a group can be embedded into the wreath product of a factor group by a normal subgroup and also proved that a factor group can be embedded inside a wreath product and the wreath product of a factor group by a factor group can be embedded into a group. We further showed that when the abstract group in the Universal Embedding Theorem is a p-group, cyclic and simple, the embedding becomes an isomorphism. Examples were given to justify the results.

General Cyclic Orthogonal Double Covers of Finite Regular Circulant Graphs

An orthogonal double cover (ODC) of a graph H is a collection of subgraphs (pages) of H, so that they cover every edge of H twice and the intersection of any two of them contains exactly one edge. An ODC G of H is cyclic (CODC) if the cyclic group of order is a subgroup of the automorphism group of G. In this paper, we introduce a general orthogonal labelling for CODC of circulant graphs and construct CODC by certain classes of graphs such as complete bipartite graph, the union of the co-cycles graph with a star, the center vertex of which, belongs to the co-cycles graph and graphs that are connected by a one vertex.

4‐SCHGDD的存在条件及组合构造

利用CHDM与SCHGDD等价,研究型为(n,m t)的4-SCHGDD的存在条件及应用。研究得到n=4时,型为(4,m t)的4-SCHGDD的存在条件;利用递归构造结合4-SCGDD的存在条件得出n≠4时,型为(n,m t)的4-SCHGDD的存在条件;最后应用结果得出AM-OPPSP/WP 3-D(u×v×w,4,1)-OOC存在的充分条件。

极大类2群的4次中心扩张

针对循环群被极大类群中心扩张的问题,在极大类2群2次中心扩张的基础上,继续探讨不同阶的循环群被极大类2群中心扩张的情况.设N为4阶循环群,H为极大类2群,G为N被H的中心扩张,给出此时G的分类.