Related papers: A new upper bound for the cross number of finite A…

We produce new short laws in two variables valid in finite groups of Lie type. Our result improves upon results of Kozma and the second named author, and is sharp up to logarithmic factors, for all families except possibly the Suzuki…

We comment on the paper "Extremal Cayley digraphs of finite Abelian groups" [Intercon. Networks 12 (2011), no. 1-2, 125--135]. In particular, we give some counterexamples to the results presented there, and provide a correct result for…

In this note we provide some counterexamples for the conjecture of Moret\'{o} on finite simple groups, which says that any finite simple group $G$ can determined in terms of its order $|G|$ and the number of elements of order $p$, where $p$…

We give new bounds and asymptotic estimates on the largest Kronecker and induced multiplicities of finite groups. The results apply to large simple groups of Lie type and other groups with few conjugacy classes.

We consider abelain subgroups of small index in finite groups. More generally, we consider subgroups such that the product of their index by the index of their centralizer is small.

It is well-known that pythagorean triples can be represented by points of the unit circle with rational coordinates. These points form an abelian group, and we describe its structure. This structural description yields, almost immediately,…

The aim of this paper is to give upper bounds for the Euclidean minima of abelian fields of odd prime conductor. In particular, these bounds imply Minkowski's conjecture for totally real number fields of conductor p^r, where p is an odd…

In this paper we obtain significant bounds for the number of maximal subgroups of a given index of a finite group. These results allow us to give new bounds for the number of random generators needed to generate a finite $d$-generated group…

After $2$-crossing-critical graphs were characterized in 2016, their most general subfamily, large $3$-connected $2$-crossing-critical graphs, has attracted separate attention. This paper presents sharp upper and lower bounds for their…

The theory of Frobenius groups with Frobenius complements of even order largely reduces to tractable algebraic number theory. If we consider only Frobenius complements with an upper bound $s$ on the number of distinct primes dividing the…

We establish an explicit lower bound for the N\'eron-Tate height on elliptic curves with complex multiplication, for nontorsion points defined over the maximal abelian extension of a number field. Building on a strategy developed by…

This paper develops some general results about actions of finite groups on (infinite) abelian groups in the finite Morley rank category. They are linked to a range of problems on groups of finite Morley rank discussed in [16]. Crucially,…

We show that, for every transitive group $G$ of degree $n\ge 2$, the largest abelian quotient of $G$ has cardinality at most $4^{n/\sqrt{\log_2 n}}$. This gives a positive answer to a 1989 outstanding question of L\'aszl\'o Kov\'acs and…

We show that if a finite, large enough subset A of an arbitrary abelian group satisfies the small doubling condition |A + A| < (log |A|)^{1 - epsilon} |A|, then A must contain a three-term arithmetic progression whose terms are not all…

In two earlier articles, we proved that, if the Hodge conjecture is true for ALL CM abelian varieties over the complex numbers, then both the Tate conjecture and the standard conjectures are true for abelian varieties over finite fields.…

We conjecture that every infinite group $G$ can be partitioned into countably many cells $G=\bigcup_{n\in\omega}A_n$ such that $cov(A_nA_n^{-1})=|G|$ for each $n\in\omega$. Here $cov(A)=\min\{|X|:X\subseteq G, G=XA\}$. We confirm this…

The {\it crossing number} of a graph $G$ is the minimum number of pairwise intersections of edges in a drawing of $G$. Motivated by the recent work [Faria, L., Figueiredo, C.M.H. de, Sykora, O., Vrt'o, I.: An improved upper bound on the…

In a recent paper, Gabriel Navarro and Pham Huu Tiep show that the so-called Alperin Weight Conjecture can be verified via the Classification of the Finite Simple Groups, provided any simple group fulfills a very precise list of conditions.…

Let $G=A_n$, a finite alternating group. We study the commuting graph of $G$ and establish, for all possible values of $n$ barring $13, 14, 17$ and $19$, whether or not the independence number is equal to the clique-covering number.

In the degree-diameter problem for Abelian Cayley and circulant graphs of diameter 2 and arbitrary degree d there is a wide gap between the best lower and upper bounds valid for all d, being quadratic functions with leading coefficient 1/4…