Related papers: Gauss Sums on $GL_2(\mathbb{Z}/p^l\mathbb{Z})$

We consider the groups of regular circulant matrices over finite fields and integer residue class rings. In both cases we present a formula for the order of these groups. We also make a first step towards finding the algebraic structure of…

The classification of the finite subgroups of $\mathrm{GL}_n(\mathbb{C})$ and $\mathrm{PGL}_n(\mathbb{C})$ is a classical problem in the field of finite group theory, dating back to the late 19th century with authors like Klein, Jordan,…

A linear group G on a finite vector space V, (that is, a subgroup of GL(V)) is called (1/2)-transitive if all the G-orbits on the set of nonzero vectors have the same size. We complete the classification of all the (1/2)-transitive linear…

In his notebooks, Gauss recorded various calculations with "infinite congruences". These infinite congruences are p-adic numbers; Gauss computes a square root of $5$ in the $11$-adic integers in order to find an $11$-adic approximation to a…

We study the distribution of the Galois group of a random $q$-additive polynomial over a rational function field: For $q$ a power of a prime $p$, let $f=X^{q^n}+a_{n-1}X^{q^{n-1}}+\ldots+a_1X^q+a_0X$ be a random polynomial chosen uniformly…

In this article, we offer group-theoretic, field-theoretic, and topological interpretations of the Gaussian binomial coefficients and their sum. For a finite $p$-group $G$ of rank $n$, we show that the Gaussian binomial coefficient…

Assuming the Generalized Riemann Hypothesis, the authors study when a character sum over all n <= x is o(x); they show that this holds if log x / log log q -> infinity and q -> infinity (q is the size of the finite field).

We use hypergeometric sheaves on $G_m/F_q$, which are particular sorts of rigid local systems, to construct explicit local systems whose arithmetic and geometric monodromy groups are the finite general linear groups $GL_n(q)$ for any $n \ge…

We study a characteristic subgroup of finitely generated groups, consisting of elements with uniform upper bound for word-lengths. For a group $G$, we denote this subgroup by $G_{bound}$. We give sufficient criteria for triviality and…

In this paper we have estimated one multi-variable character sum \begin{align*} \sum_{a=2}^{p-2}\sum_{b=1}^{p-1}\sum_{c=2}^{p-2}\sum_{d=1}^{p-1}\left(\frac{a^2-b^2}{p}\right)\left(\frac{b^2-1}{p}\right)…

Let $P\in\mathbb Q[t,x]$ be a polynomial in two variables with rational coefficients, and let $G$ be the Galois group of $P$ over the field $\mathbb Q(t)$. It follows from Hilbert's Irreducibility Theorem that for most rational numbers $c$…

We are interested in determining the bound of the average of the degrees of the irreducible characters whose degrees are not divisible by some prime $p$ that guarantees a finite group $G$ of odd order is $p$-nilpotent. We find a bound that…

We compute the first explicit polynomials with Galois groups $G=P\Gamma L_3(4)$, $PGL_3(4)$, $PSL_3(4)$ and $PSL_5(2)$ over $\mathbb{Q}(t)$. Furthermore we compute the first examples of totally real polynomials with Galois groups…

If G is a GGS-group defined over a p-adic tree, where p is an odd prime, we calculate the order of the congruence quotients $G_n=G/\Stab_G(n)$ for every n. If G is defined by the vector $e=(e_1,...,e_{p-1})\in\F_p^{p-1}$, the determination…

In this article, we determine all inequivalent irreducible rational matrix representations of groups of order $p^5$, where $p$ is an odd prime. We also derive combinatorial formulations for the Wedderburn decomposition of rational group…

We compute the characters of real irreducible representations of SL(2,q), the special linear group on q letters, for an odd prime $q$. Moreover, we give the dimensions of these irreducible representations under the actions of cyclic…

Let $G$ denote the projective special linear group $\text{PSL}(2,q)$, for a prime power $q$. It is shown that a finite 2-subgroup of the group $V(\mathbb{Z}G)$ of augmentation 1 units in the integral group ring $\mathbb{Z}G$ of $G$ is…

In positive characteristic, nearly all Picard-Vessiot extensions are inseparable over some intermediate iterative differential extensions. In the Galois correspondence, these intermediate fields correspond to nonreduced subgroup schemes of…

We study the ramification groups of finite Galois extensions $L/K$ of a complete discrete valuation field $K$ of equal characteristic $p>0$ with perfect residue field and Galois group isomorphic to the group of unitriangular matrices…

Let $p$ be a prime and $F$ be a finite field of characteristic $p$. Suppose that $FG$ is the group algebra of the finite $p$-group $G$ over the field $F$. Let $V(FG)$ denote the group of normalized units in $FG$ and let $V_*(FG)$ denote the…