Let α1,…,αr be algebraic numbers in a number field K generating subgroup of rank r K×. We investigate under GRH the primes p such that each orders (αimodp) lies given arithmetic progression associated to αi. also study for which index is fixed integer or set integers i. An additional condition on Frobenius conjugacy class may considered. Such results are generalizations theorem Ziegler from 200...

Random walk on the chambers of hyperplanes arrangements is used to de ne a family of card shu ing measuresMW;x for a nite Coxeter group W and real x 6= 0. By algebraic group theory, there is a map from the semisimple orbits of the adjoint action of a nite group of Lie type on its Lie algebra to the conjugacy classes of the Weyl group. Choosing such a semisimple orbit uniformly at random thereby...

Braid cryptosystem was proposed in CRYPTO 2000 as an alternate public-key cryptosystem. The security of this system is based upon the conjugacy problem in braid groups. Since then, there have been several attempts to break the braid cryptosystem by solving the conjugacy problem in braid groups. In this paper, we first survey all the major attacks on the braid cryptosystem and conclude that the ...

Abstract. Let SL(2, q) be the group of 2× 2 matrices with determinant one over a finite field F of size q. We prove that if q is even, then the product of any two noncentral conjugacy classes of SL(2, q) is the union of at least q − 1 distinct conjugacy classes of SL(2, q). On the other hand, if q > 3 is odd, then the product of any two noncentral conjugacy classes of SL(2, q) is the union of a...

The cycling operation endows the super summit set Sx of any element x of a Garside group G with the structure of a directed graph Γx. We establish that the subset Ux of Sx consisting of the circuits of Γx can be used instead of Sx for deciding conjugacy to x in G, yielding a faster and more practical solution to the conjugacy problem for Garside groups. Moreover, we present a probabilistic appr...

In this paper, a new family of Dai-Liao–type conjugate gradient methods are proposed for unconstrained optimization problem. the methods, modified secant equation used in [H. Yabe and M. Takano, Comput. Optim. Appl. 28 (2004) 203–225] is considered Dai Liao’s conjugacy condition. Under some certain assumptions, we show that our globally convergent general functions with strong Wolfe line search...

We show that under some simple conditions a topological conjugacy h between two holomorphic self-maps f1 and f2 of complexn-dimensional projective space Pn lifts canonically to a topological conjugacy H between the two corresponding polynomial self-maps of Cn+1, and this conjugacy relates the two Green functions of f1 and f2. These conjugacies are interesting because their geometry is not inher...

Belinskaya's theorem states that given an ergodic measure-preserving transformation, any other transformation with the same orbits and $\mathrm{L}^1$ cocycle must be flip-conjugate to it. Our main result shows this is optimal: for all $p<1$ integrability condition on cannot relaxed being in $\mathrm{L}^p$. This also allows us answer a question of Kerr Li: transformations, Shannon orbit equivale...

emph{The (undirected) power graph on the conjugacy classes} $mathcal{P_C}(G)$ of a group $G$ is a simple graph in which the vertices are the conjugacy classes of $G$ and two distinct vertices $C$ and $C'$ are adjacent in $mathcal{P_C}(G)$ if one is a subset of a power of the other. In this paper, we describe groups whose associated graphs are $k$-regular for $k=5,6$.

In this paper, we consider the conjugacy problem for HNNextensions of groups with solvable conjugacy problem for which the associated subgroups are cyclic. An example of such a group with unsolvable conjugacy problem is constructed. A similar construction is given for free products with amalgamation.