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We study cubic graphical regular representations of the finite simple groups $PSL_2(q)$. It is shown that such graphical regular representations exist if and only if $q\neq7$, and the generating set must consist of three involutions.

In this note we present the Stiefel-Whitney classes (SWCs) for orthogonal representations of several finite groups of Lie type, namely for $G=\text{SL}(2,q),$ $\text{SL}(3,q),$ $\text{Sp}(4,q)$, and $\text{Sp}(6,q)$, with $q$ odd. We also…

Let (k1,k2,k3,k4) be a quartet of cyclic cubic number fields sharing a common conductor c=pqr divisible by exactly three prime(power)s p,q,r. For those components k of the quartet whose 3-class group Cl(3,k) = Z/3Z x Z/3Z is elementary…

In this paper we prove that the unitary groups $SU_n(q^2)$ are $(2,3)$-generated for any prime power $q$ and any integer $n\geq 8$. By previous results this implies that, if $n\geq 3$, the groups $SU_n(q^2)$ and $PSU_n(q^2)$ are…

Using generating functions, we enumerate regular semisimple conjugacy classes in the finite classical groups. For the general linear, unitary, and symplectic groups this gives a different approach to known results; for the special…

We consider three families of groups: the Bianchi groups SL(2,O) where O is the ring of integers of an imaginary, quadratic field; the groups SL*(2,O) where O is a *-order of a definite, rational quaternion algebra with an orthogonal…

The differential and variational calculus on the $SL_{q}(2,R)$ group is constructed. The spontaneous breaking symmetry in the WZNW model with $SL_{q}(2,R)$ quantum group symmetry and in the $\sigma$-models with ${SL_{q}(2,R)/U_{h}(1)}$…

A two-parametric deformation of U[sl(2)] and its representations are considered. This newly introduced two-parametric quantum group denoted as $U_{pq}[sl(2)]$ admits a class of infinite-dimensional representations which have no classical…

I introduce compact quantum group extensions associated with the $q$-deformations of the classical compact groups $USp(2n)$, $O(n,\mathbb{R})$ and $SO(2n,\mathbb{R})$. Motivated by the relationship between $SU_q(n)$ and $U_q(n)$, I study…

We show that a surface group of high genus contained in a classical simple Lie group can be deformed to become Zariski dense, unless the Lie group is $SU(p,q)$ (resp. $SO^* (2n)$, $n$ odd) and the surface group is maximal in some…

We show that the Grothendieck-Teichm\"uller group of $PSL(2, q)$, or more precisely the group $GT_1(PSL(2, q))$ as previously defined by the author, is the product of an elementary abelian 2-group and several copies of the dihedral group of…

Let $\mathcal{S}$ be a finite thick generalized quadrangle, and suppose that $G$ is an automorphism group of $\mathcal{S}$. If $G$ acts primitively on both the points and lines of $\mathcal{S}$, then it is known that $G$ must be almost…

In this article we extend our previous results for the orthogonal group, $SO(2,4)$, to its homomorphic group $SU(2,2)$. Here we present a closed, finite formula for the exponential of a $4\times 4$ traceless matrix, which can be viewed as…

The expressions for the $\hat{R}$--matrices for the quantum groups SO$_{q^2}$(5) and SO$_q$(6) in terms of the $\hat{R}$--matrices for Sp$_q$(2) and SL$_q$(4) are found, and the local isomorphisms of the corresponding quantum groups are…

The two-parametric quantum superalgebra $U_{pq}[gl(2/2)]$ and its representations are considered. All finite-dimensional irreducible representations of this quantum superalgebra can be constructed and classified into typical and nontypical…

In this paper, we consider the order m k-automorphisms of SL(2,k). We first characterize the forms that order m k-automorphisms of SL(2,k) take and then we simple conditions on matrices A and B, involving eigenvalues and the field that the…

In the present paper we construct all typical finite-dimensional representations of the quantum Lie superalgebra $U_{q}[gl(2/2)]$ at generic deformation parameter $q$. As in the non-deformed case the finite-dimensional…

In this paper we consider the restriction of a unitary irreducible representation of type $A_{\mathfrak q}(\lambda)$ of $GL(4,{\mathbb R})$ to reductive subgroups $H$ which are the fixpoint sets of an involution. We obtain a formula for the…

In this thesis, we develop algorithms similar to the Gaussian elimination algorithm in symplectic and split orthogonal similitude groups. As an application to this algorithm, we compute the spinor norm for split orthogonal groups. Also, we…

We classify, up to conjugacy, the finite subgroups of PGL(2,K) of order prime to char(K).