相关论文: Diassociativity in Conjugacy Closed Loops

Let ${\cal C}$ be a nonempty class of finite groups closed under taking subgroups, homomorphic images and extensions. A subgroup $H$ of an abstract residually ${\cal C}$ group $R$ is said to be conjugacy ${\cal C}$-distinguished if whenever…

A group is nested if the centers of the irreducible characters form a chain. In this paper, we will show that there is a set of subgroups associated with the conjugacy classes of group so that a group is nested if and only if these…

In this paper we study conjugacy and subgroup separability properties in the class of nilpotent $\mathbb{Q}[x]$-powered groups. Many of the techniques used to study these properties in the context of ordinary nilpotent groups carry over…

Let $\F$ be an algebraically closed field. Let $\V$ be a vector space equipped with a non-degenerate symmetric or symplectic bilinear form $B$ over $\F$. Suppose the characteristic of $\F$ is \emph{large}, i.e. either zero or greater than…

Let $G$ be a finite group. Let $k(G)$ denote the number of conjugacy classes of $G$ and let $m(G)$ denote the least positive integer $n$ such that the union of any $n$ distinct non-trivial conjugacy classes of $G$ together with the identity…

Let $G$ be a finite group and $N$ a normal subgroup of $G$. We determine the structure of $N$ when the diameter of the graph associated to the $G$-conjugacy classes contained in $N$ is as large as possible, that is, is equal to three.

Parastrophes (conjugates) of a quasigroup can be divided into separate classes containing isotopic parastrophes. We prove that the number of such classes is always 1, 2, 3 or 6. Next we characterize quasigroups having a fixed number of such…

Many results have been established that show how the number of conjugacy classes appearing in the product of classes affect the structure of a finite group. The aim of this paper is to show several results about solvability concerning the…

The Divisibility Graph of a finite group $G$ has vertex set the set of conjugacy class lengths of non-central elements in $G$ and two vertices are connected by an edge if one divides the other. We determine the connected components of the…

Let $G$ be a finite group and let $N$ be a normal subgroup of $G$. We attach to $N$ two graphs ${\Gamma}_G(N)$ and ${\Gamma}^{\ast}_G(N)$ related to the conjugacy classes of $G$ contained in $N$ and to the set of primes dividing the sizes…

In this paper, we study nilpotent $\mathbb{Q}$$[x]$-powered groups that satisfy the following property: For some set of primes $\omega$ in $\mathbb{Q}$$[x]$, every $\omega '$-isolated $\mathbb{Q}$$[x]$-subgroup in some family of its…

This note presents the study of the conjugacy classes of elements of some given finite order n in the Cremona group of the plane. In particular, it is shown that the number of conjugacy classes is infinite if n is even, n=3 or n=5, and that…

Let H and K be quasiconvex subgroups of a negatively curved torsion-free group G. We give an algorithm which decides whether an element of H is conjugated in G to an element of K.

A semigroup conjugacy is an equivalence relation that equals group conjugacy when the semigroup is a group. In this note, we answer five open problems related to semigroup conjugacy. (Problem One) We say a conjugacy ~ is partition-covering…

For any conjugacy class C in G=PSL(2,q) we compute C^2 and discuss whether C contains a triple of elements whose product is 1 which generate G. Moreover, we determine which elements in G can be written as a product of two conjugate elements…

A group $G$ is twisted conjugacy separable if for every automorphism $\varphi$, distinct $\varphi$-twisted conjugacy classes can be separated in a finite quotient. Likewise, $G$ is completely twisted conjugacy separable if for any group $H$…

Let $G\in\{p,q\}^*$ be a finite group with trivial center, where $p,q\in\pi(G)$ and $p>q>5$. In the present paper it is proved that $|G|_{\{p,q\}}=|G||_{\{p,q\}}$; in particular $C_G(g)\cap C_G(h)=1$ for every $p$-element $g$ and every…

C-loops are loops satisfying the identity $x(y\cdot yz) = (xy\cdot y)z$. We develop the theory of extensions of C-loops, and characterize all nuclear extensions provided the nucleus is an abelian group. C-loops with central squares have…

There are several graphs defined on groups. Among them we consider graphs whose vertex set consists conjugacy classes of a group $G$ and adjacency is defined by properties of the elements of conjugacy classes. In particular, we consider…

It is proved that the maximum condition for subloops in a commutative Moufang loop $Q$ is equivalent with the conditions of finite generating of different subloops of the loop $Q$ and different subgroups of the multiplication group of the…