On Finite Groups with Special Conjugacy Classes

Let G be a finite group with the property that for any conjugacy class order, G has exactly two conjugacy classes which have the same order. We prove that： （1） ff a Sylow 2-subgroup of G is Abelian, then G is isomorphic to the direct product of symmetric group with order 3 and cyclic group with order 2, or G is isomorphic to the semidirect product of a cyclic group with order 3 and a cyclic group with order 4; （2） if G＇ is nilpotent, then G is a group of {2,3,5 }.

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.

Multiplicity one, local and global conjugacy

The issue of local and global conjugacy is closely related to the multiplicity one property in representation theory and the Langlands program. In this article we give first families of connected instances for SO2N where the multiplicity one fails in both aspects of representation theory and automorphic forms with certain assumptions on the Langlands functoriality.

Curve selection Lemma for semianalytic sets and conjugacy classes of finite order in Lie groups

Using a strong version of the Curve Selection Lemma for real semianalytic sets, we prove that for an arbitrary connected Lie group G, each connected component of the set E_n(G)of all elements of order n in G is a conjugacy class in G. In particular, all conjugacy classes of finite order in G are closed. Some properties of connected components of E_n(G) are also given.

Classification of C^1 Diffeomorphisms of the Real Line Under C^1 Conjugacy

This note will give some classification results for local and global C^1 diffeomorphisms of R under C^1 conjugacy. For γ=1, 2,…, ∞, ω, let D~γ(0)={f: R→R is C~γ; f’(x)】0, x∈R; and f has 0 asits unique fixed point} (local diffeomorphisms of R). In Refs. [1] and [2], we have considered the existence of smooth embedding flows and the other related problems for f∈D~γ(0). As a result, we prove that the following classifications have some numerical invariants

C^(0) conjugacy between PM functions with nonmonotonicity height equal to 1

In this paper,a topological classification of piecewise monotone functions(abbreviated as PM functions)with nonmonotonicity height equal to 1 which are strictly increasing on their characteristic intervals and have finitely many fixed points is presented.

On a Graph Related to Subgroups Generated by a Conjugacy Class in Finite Groups

Let G be a group,and let →/ΓC(G) be a digraph whose vertices are the nontrivial conjugacy classes of G and there is an arc from a vertex C to a vertex C' if and only if C≠C' and(C)■(C').In this paper,we characterize finite groups G whose associated digraphs →/ΓC(G) are oriented trees.

On Conjugacy Class Graph of Normal Subgroup

Let G be a finite group and N a normal subgroup of G.Denote by Γ_(G)(N)the graph whose vertices are all distinct G-conjugacy class sizes of non-central elements in N,and two vertices of Γ_(G)(N)are adjacent if and only if they are not coprime numbers.We prove that if the center Z(N)=Z(G)∩N and Γ_(G)(N)is k-regular for k≥1,then either a section of Nis a quasi-Frobenius group or Γ_(G)(N)is a complete graph with k+1 vertices.

An Arithmetical Condition on the Sizes of Conjugacy Classes of a Finite Group

An element x of a finite group G is said to be primary if the order of x is a prime power.We define csp2(G)as follows:if|x^(G)|is a prime power for every primary element x of G,where x^(G) is the conjugacy class of x in G,then csp2(G)=0;if there exists a primary element x in G such that|x^(G)|is divisible by at least two distinct primes,then csp2(G)=max{|x^(G)||x∈Gis primary,|x^(G)|is divisible by at least two distinct primes}.In this paper we discuss the influence of the number csp2(G)on the structure of G.

An Adaptive Three-Term Conjugate Gradient Method with Sufficient Descent Condition and Conjugacy Condition

In this paper,an adaptive three-term conjugate gradient method is proposed for solving unconstrained problems,which generates sufficient descent directions at each iteration.Different from the existent methods,a dynamical adjustment between Hestenes–Stiefel and Dai–Liao conjugacy conditions in our proposed method is developed.Under mild condition,we show that the proposed method converges globally.Numerical experimentation with the new method indicates that it efficiently solves the test problems and therefore is promising.

Authenticated Group Key Agreement Protocol Based on Twist Conjugacy Problem in Near-Rings

Nowadays some promising authenticated group key agreement protocols are constructed on braid groups, dynamic groups, pairings and bilinear pairings. Hence the non-abelian structure has attracted cryptographers to construct public-key cryptographic protocols. In this article, we propose a new authenticated group key agreement protocol which works in non-abelian near-rings. We have proved that our protocol meets the security attributes under the assumption that the twist conjugacy search problem（TCSP） is hard in near-ring.

A Result on Character Degrees and Conjugacy Class Sizes in Finite Groups

Let G be a finite group and π be a set of primes including at least two elements. We write cd（G） and cs（G） to denote the set of complex irreducible character degrees and conjugacy class sizes of G , respectively, and write π（m）to denote the set of all prime divisors of a positive integer m . For any 1≠m∈cd（G） and 1≠m∈cs（G）, in this note, we shall present the corresponding group structures of finite group G in the case π（m）=π , respectively, which generalizes the result of finite groups with character degrees of two distinct primes. Furthermore, we shall see that the influence of the two sets on the group structure is analogous.

On the Smoothness of Conjugacy for Circle Covering Maps

A conjugaey between C1+αΣ-hyperbolic circle covering maps is C1+αif it has a positive derivative on some point in the Σ-set.

NEW HYBRID CONJUGATE GRADIENT METHOD AS A CONVEX COMBINATION OF LS AND FR METHODS

In this paper, we present a new hybrid conjugate gradient algorithm for unconstrained optimization. This method is a convex combination of Liu-Storey conjugate gradient method and Fletcher-Reeves conjugate gradient method. We also prove that the search direction of any hybrid conjugate gradient method, which is a convex combination of two conjugate gradient methods, satisfies the famous D-L conjugacy condition and in the same time accords with the Newton direction with the suitable condition. Furthermore, this property doesn't depend on any line search. Next, we also prove that, moduling the value of the parameter t,the Newton direction condition is equivalent to Dai-Liao conjugacy condition.The strong Wolfe line search conditions are used.The global convergence of this new method is proved.Numerical comparisons show that the present hybrid conjugate gradient algorithm is the efficient one.

REPELLERS FOR MULTIFUNCTIONS OF SEMI-BORNOLOGICAL SPACES

In this article the notion of repeller for multifunctions from the viewpoints of semi-bornologicul spaces is considered. The concept of lower semi-continuous multifunctions is extended by the use of semi-bornological spaces. Semi-bornological vector spaces are studied. The notion of conjugacy for semi-bornological multifunctions is considered. The persistence of repeller under conjugate relation is proved.

New type of conjugate gradient algorithms for unconstrained optimization problems

Two new formulaes of the main parameter βk of the conjugate gradient method are presented, which respectively can be seen as the modifications of method HS and PRP. In comparison with classic conjugate gradient methods, the new methods take both available gradient and function value information. Furthermore, their modifications are proposed. These methods are shown to be global convergent under some assumptions. Numerical results are also reported.

COSET DIAGRAMS FOR A HOMOMORPHIC IMAGE OFΔ(3,3,k)

Let q be a prime power. By PL（Fq） the authors mean a projective line over the finite field Fq with the additional point ∞. In this article, the authors parametrize the conjugacy classes of nondegenerate homomorphisms which represent actions of △（3, 3, k） = （u, v： u^3 = v^3 = （uv）^k = 1〉on PL（Fq）, where q ≡ ±1（modk）. Also, for various values of k, they find the conditions for the existence of coset diagrams depicting the permutation actions of △（3, 3, k） on PL（Fq）. The conditions are polynomials with integer coefficients and the diagrams are such that every vertex in them is fixed by （u^-v^-）^k. In this way, they get △（3, 3, k） as permutation groups on PL（Fq）.

Groups Having Elements Conjugate to Their Squares and Connections with Dynamical Systems

In recent years, dynamical systems which are conjugate to their squares have been studied in ergodic theory. In this paper we study the consequences of groups having elements which are conjugate to their squares and consider examples arising from topological dynamics and more general dynamical

The Class Number of Derived Subgroups and the Structure of Camina Groups

A finite group G is called a Camina group if G has a proper normal subgroup N such that gN is precisely a conjugacy class of G for any g ∈E G - N. In this paper, the structure of a Camina group G is determined when N is a union of 2, 3 or 4 conjugacy classes of G.

A Note on the Prime Spectrums of Character Rings of Finite Groups

Let G be a finite group with order g and S be a subring of the algebraic number field which contains the integral extension over Z generated by a g-th primitive root co of unity, and R（G） be the character ring of G. The prime spectrum of the commutative ring S×Z R（G） iv denoted by Spec（S×Z R（G）） and set π={p｜p is a rational prime number such that p^-1 S}. We prove that when G is a regroup, a π＇-group, or a finite Abelian group, the number of the connetted components of Spec（ S×Z R （G） ） coincides with the number of the π-regular classes in G,