Related papers: Primitive elements in $p$-divisible groups

A universal Gr\"obner basis of an ideal is the union of all its reduced Gr\"obner bases. It is contained in the Graver basis, the set of all primitive elements. Obtaining an explicit description of either of these sets, or even a sharp…

Let $G$ be a finite group and $p$ be a prime. We study the kernel of the map, between the Burnside ring of $G$ and the Grothendieck ring of $\mathbb{F}_p[G]$-modules, taking a $G$-set to its associated permutation module. We are able, for…

Building upon the author's previous work on primitivity testing of finite nilpotent linear groups over fields of characteristic zero, we describe precisely those finite nilpotent groups which arise as primitive linear groups over a given…

Under GRH, any element in the multiplicative group of a number field $K$ that is globally primitive (i.e., not a perfect power in $K^*$) is a primitive root modulo a set of primes of $K$ of positive density. For elliptic curves $E/K$ that…

A primitive prime divisor of an element a_n of a sequence (a_1,a_2,a_3,...) is a prime P that divides a_n, but does not divide a_m for all m < n. The Zsigmondy set Z of the sequence is the set of n such that a_n has no primitive prime…

Explicit generating sets are found for all primitive ideals in the generic quantized coordinate rings of the 3x3 special and general linear groups over an arbitrary algebraically closed field. (Previously, generators were only known up to…

Let $\mathbb{F}_{q^n}$ be a finite field with $q^n$ elements and $r$ be a positive divisor of $q^n-1$. An element $\alpha \in \mathbb{F}_{q^n}^*$ is called $r$-primitive if its multiplicative order is $(q^n-1)/r$. Also, $\alpha \in…

The structure of finite and locally finite groups in which every element has prime power order (CP-groups) is well known. In this paper we note that the combination of our earlier results with the available information on the structure of…

We characterize the finite groups of minimal order that admit an irreducible complex character of degree $p$ or $p^2$, where $p$ is a prime.

In this paper we generalize the results of Sharma, Awasthi and Gupta (see \cite{SAG}). We work over a field of any characteristic with $q = p^k$ elements and we give a sufficient condition for the existence of a primitive element $\alpha…

Let $P$ be a non-torsion point on the elliptic curve $E_{a}: y^{2}=x^{3}+ax$. We show that if $a$ is fourth-power-free and either $n>2$ is even or $n>1$ is odd with $x(P)<0$ or $x(P)$ a perfect square, then the $n$-th element of the…

Silverman proved the analogue of Zsigmondy's Theorem for elliptic divisibility sequences. For elliptic curves in global minimal form, it seems likely this result is true in a uniform manner. We present such a result for certain infinite…

We consider 2-local geometries and other subgroup complexes for sporadic simple groups. For six groups, the fixed point set of a noncentral involution is shown to be equivariantly homotopy equivalent to a standard geometry for the component…

We study the primitive divisors of the terms of $(\Delta_n)_{n \geq 1}$, where $\Delta_n=N_{K/ \mathbb{Q}}(u^n-1)$ for $K$ a real quadratic field, and $u>1$ a unit element of its ring of integers. The methods used allow us to find the terms…

In this paper, the notion of a uniformly distributed systems of elements on the variety of metabelian Lie algebras is introduced. This notion is analogous to one of a measure preserving systems of elements on group varieties. As the main…

Fundamental elements are certain special elements of affine Weyl groups introduced by Gortz, Haines, Kottwitz and Reuman. They play an important role in the study of affine Deligne-Lusztig varieties. In this paper, we obtain…

In this paper we continue the study of prime graphs of finite solvable groups. The prime graph, or Gruenberg-Kegel graph, of a finite group G has vertices consisting of the prime divisors of the order of G and an edge from primes p to q if…

This paper is a continuation of our previous works where we study maps from $X_0(N)$, $N \ge 1$, into $\mathbb P^2$ constructed via modular forms of the same weight and criteria that such a map is birational (see [12]). In the present paper…

Polynomials and elements over finite fields exhibit closely related algebraic structures, and many properties defined for elements extend naturally to polynomials. The concepts of order and $\mathbb{F}_q$-Order for elements have been…

The primitive curves are the multiple curves that can be locally embedded in smooth surfaces (we will always suppose that the associated reduced curves are smooth). These curves have been defined and studied by C. Banica and O.Forster in…