Related papers: Products of conjugacy classes in $\text{SL}_2(k)$ …

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…

In this note, we classify the conjugacy classes of $\widetilde{\mathrm{SL}}_2(\mathbb{R})$, the universal covering group of $\mathrm{PSL}_2(\mathbb{R})$. For any non-central element $\alpha \in \widetilde{\mathrm{SL}}_2(\mathbb{R})$, we…

We compute the product of any $n$-tuple of conjugacy classes in $SL_2(R)$

For the finite groups GU(3), SU(3), GL(3), SL(3) over a finite field we solve the class product problem, i.e., we give a complete list of $m$-tuples of conjugacy classes whose product does not contain the identity matrix.

We call a conjugacy class of the symplectic group Sp$(2n, K)$ over a field $K$ strictly hyperbolic if its minimal polynomial is of the form $q(x) q^*(x)$, where the polynomial $q(x)$ is prime to its reciprocal $q^*(x) := x^n q(x^{-1})$. It…

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…

It is shown that the product of two nonscalar conjugacy classes of the special linear group SL$(n,K)$ contains matrices of arbitrary trace if $n \ge 4$ and $K$ is an abitrary field or $n=3$ and $K$ is finite.

Let GL(n,q) be the group of nxn invertible matrices over a field with q elements, and SL(n,q) be the group of nxn matrices with determinant 1 over a field with q elements. We prove that the product of any two non-central conjugacy classes…

We obtain a complete description of collections of n conjugacy classes in SU(2) with the property that the multiplication map from the product of these n conjugacy classes to SU(2) is surjective. The basic instrument is a characterization…

We first describe, over a field K of characteristic different from 2, the orbits for the adjoint actions of the Lie groups PGL(2, K) and PSL(2, K) on their Lie algebra sl(2, K). While the former are well known, the latter lead to the…

We summarize several results about non-simplicity, solvability and normal structure of finite groups related to the number of conjugacy classes appearing in the product or the power of conjugacy classes. We also collect some problems that…

We classify the real and strongly real conjugacy classes in $GL_n(q)$, $SL_n(q)$, $PGL_n(q)$, $PSL_n(q)$, and all quasi-simple covers of $PSL_n(q)$. In each case we give a formula for the number of real, and the number of strongly real,…

Let G be a simple algebraic group over an algebraically closed field k. We classify the spherical conjugacy classes of G.

The aim of this work is to study finite dimensional representations of the Lie superalgebra psl(2|2) and their tensor products. In particular, we shall decompose all tensor products involving typical (long) and atypical (short)…

We describe the subracks of the conjugacy classes of $\mathrm{PSL}(2,q)$ based on Dickson's theorem on subgroups of $\mathrm{PSL}(2,q)$. All minimal non-abelian subracks of $\mathrm{PSL}(2,q)$ are determined. Further, we provide a general…

In this paper, we prove that if G is a finite simple group with the same-size conjugacy class set U(G) = U(PSL(2, 11)), then G is isomorphic to PSL(2, 11).

We classify pairs of conjugacy classes in almost simple algebraic groups whose product consists of finitely many classes. This leads to several interesting families of examples which are related to a generalization of the Baer--Suzuki…

Let A_n be the alternating group on n letters. For n>5, we describe the elements alpha, beta in A_n when alpha^{A_n} beta^{A_n} is the union of at most four distinct conjugacy classes.

We prove several results on products of conjugacy classes in finite simple groups. The first result is that there always exists a uniform generating triple. This result and other ideas are used to solve a 1966 conjecture of Peter Neumann…

For a simple algebraic group $G$ over an algebraically closed field, we study products of normal subsets. For this we mark the nodes of the Dynkin diagram of $G$. We use two types of labels, a binary marking and a labeling with non-negative…