Related papers: Derived Length and Products of Conjugacy Classes

We study the properties of the relative derived category $D_{\mathscr{C}}^{b}$($\mathscr{A}$) of an abelian category $\mathscr{A}$ relative to a full and additive subcategory $\mathscr{C}$. In particular, when $\mathscr{A}=A{\text -}\mod$…

Given an infinite, compact, monothetic group $G$ we study decompositions and structure of unbounded derivations in a crossed product C$^*$-algebra $C(G)\rtimes\Z$ obtained from a translation on $G$ by a generator of a dense cyclic subgroup.…

Let $S$ be a semigroup. The elements $a,b\in S$ are called primarily conjugate if $a=xy$ and $b=yx$ for certain $x,y\in S$. The relation of conjugacy is defined as the transitive closure of the relation of primary conjugacy. In the case…

Let $G$ be a finite group, $N$ a normal subgroup of $G$ and $x\in G-N$. We discuss when the coset $Nx$ is contained in the union of two conjugacy classes, $K$ and $D$, of $G$. We show that $N$ need not be solvable, and can even be…

We study groups $G$ where the $\varphi$-conjugacy class $[e]_{\varphi}=\{g^{-1}\varphi(g)~|~g\in G\}$ of the unit element is a subgroup of $G$ for every automorphism $\varphi$ of $G$. If $G$ has $n$ generators, then we prove that the $k$-th…

Recently there has been considerable interest in studying the length and the depth of finite groups, algebraic groups and Lie groups. In this paper we introduce and study similar notions for algebras. Let $k$ be a field and let $A$ be an…

Let SL(2,q) be the group of 2X2 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…

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…

Let $H$ be a Krull monoid with class group $G$ such that every class contains a prime divisor. Then every nonunit $a \in H$ can be written as a finite product of irreducible elements. If $a=u\_1 \cdot \ldots \cdot u\_k$, with irreducibles…

Let $G$ be a finite group. An element $g$ of $G$ is called a vanishing element if there exists an irreducible character $\chi$ of $G$ such that $\chi(g) = 0$; in this case, we say that the conjugacy class of $g$ is a vanishing conjugacy…

In this paper, we prove that all finite solvable groups satisfy the Isaacs-Seitz conjecture namely the derived lenght of a finite solvable group G is less than or equal to the number of distinct irreducible complex character degrees of G.

Let $G$ be a solvable subgroup of the group $\diff{}{n}$ of local complex analytic diffeomorphisms. Analogously as for groups of matrices we bound the solvable length of $G$ by a function of $n$. Moreover we provide the best possible bounds…

In this paper, we derive an asymptotic formula for the number of conjugacy classes of elements in a class of statistically convex-cocompact actions with contracting elements. Denote by $\mathcal C(o, n)$ (resp. $\mathcal C'(o, n)$) the set…

Let $G$ be a finite group and $\sigma_1(G)=\frac{1}{|G|}\sum_{H\leq G}\,|H|$. Under some restrictions on the number of conjugacy classes of (non-normal) maximal subgroups of $G$, we prove that if $\sigma_1(G)<\frac{117}{20}\,$, then $G$ is…

In this paper, we consider Problem 14.44 in the Kourovka notebook, which is a conjecture about the number of conjugacy classes of a finite group. While elementary, this conjecture is still open and appears to elude any straightforward…

The conjugator length function of a finitely generated group $\Gamma$ gives the optimal upper bound on the length of a shortest conjugator for any pair of conjugate elements in the ball of radius $n$ in the Cayley graph of $\Gamma$. We…

The paper develops the result of second Thomae theorem in hyperelliptic case. The main formula, called general Thomae formula, provides expressions for values at zero of the lowest non-vanishing derivatives of theta functions with singular…

In this paper we establish decomposition theorems for derivations of group rings. We provide a topological technique for studying derivations of a group ring $A[G]$ in case $G$ has finite conjugacy classes. As a result, we describe all…

Let $G$ be a finite group of order divisible by a prime $p$. The number of $p$-regular and $p'$-regular conjugacy classes of $G$ is at least $2\sqrt{p-1}$. Also, the number of $p$-rational and $p'$-rational irreducible characters of $G$ is…

Suppose that a finitely generated group $G$ is hyperbolic relative to a collection of subgroups $\mathbb{P}=\{P_1,\dots,P_m\}$. Let $H_1,H_2$ be subgroups of $G$ such that $H_1$ is relatively quasiconvex with respect to $\mathbb{P}$ and…