arXiv subjects

Sávio Ribas

Publications and source records attributed to Sávio Ribas.

17 recordsLinked to original sources

arXiv2026

On a class of combinatorial group invariants

In this paper, we explore group invariants arising from combinatorial structures associated with finite groups, including the functional graphs of power maps and the well-studied power graphs. These invariants induce equivalence relations (and hence partitions) on the set of isomorphism classes of finite groups, which we classify from the finest to the coarsest. Surprisingly, all but three of these partitions turn out to coincide; for the subclass of nilpotent groups, all but two coincide. Furthermore, we introduce a broad class of nilpotent groups and show that, within this class, all but one of these partitions agree. Our proofs draw on tools and ideas from Combinatorics and Number Theory, while requiring only basic notions from Group Theory. In particular, we construct a general framework that may prove useful in contexts similar to those considered in this paper. Finally, we propose some open questions that emerge from our results.

math.CO↗

arXiv2026

On zero-sum problems over metacyclic groups $C_n \rtimes_s C_2$

Let $G$ be a finite group. A finite collection of elements from $G$, where the order is disregarded and repetitions are allowed, is said to be a product-one sequence if its elements can be ordered such that their product in $G$ equals the identity element of $G$. Then, the Gao's constant $\mathsf E (G)$ of $G$ is the smallest integer $\ell$ such that every sequence of length at least $\ell$ has a product-one subsequence of length $|G|$. For a positive integer $n$, we denote by $C_n$ a cyclic group of order $n$. Let $G = C_n \rtimes_s C_2$ with $s^2\equiv 1\pmod n$ be a metacyclic group. The direct and inverse problems of $\mathsf E (G)$ were settled recently, except for the case that $G=C_{3n_2}\rtimes_s C_2$ with $n_2\neq 1$, $\gcd(n_2,6)=1$, $s\equiv -1 \pmod 3$, and $s\equiv 1\pmod {n_2}$. In this paper, we complete the remaining case and hence for all metacyclic groups of the form $G=C_n \rtimes C_2$, the Gao's constant and the associated inverse problem are now fully settled (see Theorem 1.2).

math.CO↗

arXiv2026

Arithmetic properties of arguments of algebraic numbers on the unit circle

An irrational number $θ$ is called Diophantine if there exist $c>0$ and $τ< \infty$ such that $\left| θ- \frac{p}{q} \right| \ge \frac{c}{q^τ}$ holds for every $(p,q) \in \mathbb{Z} \times \mathbb{N}$. In this paper, we study Diophantine and transcendence properties of some real numbers. Using lower bounds for linear forms in logarithms, we show that if $β\in \mathbb{C}$ is an algebraic number with $|β|=1$ that is not a root of unity, then $\frac{\operatorname{Arg}(β)}{2π}$ is Diophantine. We also prove that if $β= e^{iα}$ is algebraic, then $\fracαπ$ is either rational or transcendental. As a consequence, we obtain that if $n \ge 2$ is an integer and $α\in \left(0,\fracπ{2}\right)$ satisfies $n \tan α= \tan(n α)$, then $\fracα{2π}$ is both Diophantine and transcendental, and $α$ is transcendental. This extends a result of [V. Cyr, A number theoretic question arising in the geometry of plane curves and in billiard dynamics, Proc. Amer. Math. Soc. 140 (2012), no. 9, 3035--3040], which establishes that $\fracα{2π}$ is irrational.

math.NT↗

arXiv2025

On minimal product-one sequences of maximal length over the non-abelian group of order $pq$

Let $G$ be a finite group. A sequence over $G$ is a finite multiset of elements of $G$, and it is called product-one if its terms can be ordered so that their product is the identity of $G$. The large Davenport constant $\D(G)$ is the maximal length of a minimal product-one sequence, that is, a product-one sequence that cannot be partitioned into two nontrivial product-one subsequences. Let $p,q$ be odd prime numbers with $p \mid q-1$ and let $C_q \rtimes C_p$ denote the non-abelian group of order $pq$. It is known that $\D(C_q \rtimes C_p) = 2q$. In this paper, we describe all minimal product-one sequences of length $2q$ over $C_q \rtimes C_p$. As an application, we further investigate the $k$-th elasticity (and, consequently, the union of sets containing $k$) of the monoid of product-one sequences over these groups.

math.GR↗

arXiv2025

Some zero-sum problems over $\langle x,y \mid x^2 = y^{n/2}, y^n = 1, yx = xy^s \rangle$

Let $n \ge 8$ be even, and let $G = \langle x, y \mid x^2 = y^{n/2}, y^n = 1, yx = xy^s \rangle$, where $s^2 \equiv 1 \pmod n$ and $s \not\equiv \pm1 \pmod n$. In this paper, we provide the precise values of some zero-sum constants over $G$, namely the small Davenport constant, $η$-constant, Gao constant, and Erd\H os-Ginzburg-Ziv constant. In particular, the Gao's and Zhuang-Gao's Conjectures hold for $G$. We also solve the associated inverse problems when $n \equiv 0 \pmod 4$.

math.NT↗

arXiv2023

$r$-primitive $k$-normal elements in arithmetic progressions over finite fields

Let $\mathbb{F}_{q^n}$ be a finite field with $q^n$ elements. For a positive divisor $r$ of $q^n-1$, the element $α\in \mathbb{F}_{q^n}^*$ is called \textit{$r$-primitive} if its multiplicative order is $(q^n-1)/r$. Also, for a non-negative integer $k$, the element $α\in \mathbb{F}_{q^n}$ is \textit{$k$-normal} over $\mathbb{F}_q$ if $\gcd(αx^{n-1}+ α^q x^{n-2} + \ldots + α^{q^{n-2}}x + α^{q^{n-1}} , x^n-1)$ in $\mathbb{F}_{q^n}[x]$ has degree $k$. In this paper we discuss the existence of elements in arithmetic progressions $\{α, α+β, α+2β, \ldotsα+(m-1)β\} \subset \mathbb{F}_{q^n}$ with $α+(i-1)β$ being $r_i$-primitive and at least one of the elements in the arithmetic progression being $k$-normal over $\mathbb{F}_q$. We obtain asymptotic results for general $k, r_1, \dots, r_m$ and concrete results when $k = r_i = 2$ for $i \in \{1, \dots, m\}$.

math.NT↗

arXiv2023

A note on Bass' conjecture

For a finite group $G$, we denote by ${\sf d}(G)$ and by ${\sf E}(G)$, respectively, the small Davenport constant and the Gao constant of $G$. Let $C_n$ be the cyclic group of order $n$ and let $G_{m,n,s} = C_n \rtimes_s C_m$ be a metacyclic group. In [J. Bass; {\em Improving the Erdős-Ginzburg-Ziv theorem for some non-abelian groups.} J. Number Theory {\bf 126} (2007), 217-236, Conjecture 17], Bass conjectured that ${\sf d}(G_{m,n,s}) = m+n-2$ and ${\sf E}(G_{m,n,s}) = mn+m+n-2$ provided $ord_n(s) = m$. In this paper, we show that the assumption $ord_n(s) = m$ is essential and cannot be removed. Moreover, if we suppose that Bass' conjecture holds for $G_{m,n,s}$ and the $mn$-product-one free sequences of maximal length are well behaved, then Bass conjecture also holds for $G_{2m,2n,r}$, where $r^2 \equiv s \pmod n$.

math.NT↗

arXiv2022

On arithmetic progressions in finite fields

In this paper, we explore the existence of $m$-terms arithmetic progressions in $\mathbb{F}_{q^n}$ with a given common difference whose terms are all primitive elements, and at least one of them is normal. We obtain asymptotic results for $m \ge 4$ and concrete results for $m \in \{2,3\}$, where the complete list of exceptions when the common difference belongs to $\mathbb{F}_{q}^*$ is obtained. The proofs combine character sums, sieve estimates, and computational arguments using the software SageMath.

math.NT↗

arXiv2022

On the direct and inverse zero-sum problems over $C_n \rtimes_s C_2$

Let $C_n$ be the cyclic group of order $n$. In this paper, we provide the exact values of some zero-sum constants over $C_n \rtimes_s C_2$ where $s \not\equiv \pm1 \pmod n$, namely $η$-constant, Gao constant, and Erdős-Ginzburg-Ziv constant (the latter for all but a "small" family of cases). As a consequence, we prove the Gao's and Zhuang-Gao's Conjectures for groups of this form. We also solve the associated inverse problems by characterizing the structure of product-one free sequences over $C_n \rtimes_s C_2$ of maximum length.

math.NT↗

arXiv2021

The $\{1,s\}$-weighted Davenport constant in $C_n^k$

Let $G$ be a finite abelian group and let $\varnothing \neq A \subset \mathbb Z$. The $A$-weighted Davenport constant of $G$ is the smallest positive integer ${\sf D}_A(G)$ such that every sequence $x_1 \boldsymbol{\cdot} {\dots} \boldsymbol{\cdot} x_{{\sf D}_A(G)}$ over $G$ has a non-empty subsequence $(x_{j_i})_i$ such that ${\varepsilon_1} x_{j_1} + {\varepsilon_2} x_{j_2} + {\dots} + {\varepsilon_t} x_{j_t} = 0$ for some $\varepsilon_1, \varepsilon_2, {\dots}, \varepsilon_t \in A$. In this paper, we obtain both upper and lower bounds for ${\sf D}_{\{1,s\}}(C_n^k)$, where $C_n$ denotes the cyclic group of order $n$, $s^2 \equiv 1 \pmod n$ and $s \not\equiv \pm1 \pmod n$. These bounds become sharp in some "small" cases.

math.NT↗

arXiv2021

Extremal product-one free sequences over $C_n \rtimes_s C_2$

Let $G$ be a finite group multiplicatively written. The small Davenport constant of $G$ is the maximum positive integer ${\sf d}(G)$ such that there exists a sequence $S$ of length ${\sf d}(G)$ for which every subsequence of $S$ is product-one free. Let $s^2 \equiv 1 \pmod n$, where $s \not\equiv \pm1 \pmod n$. It has been proven that ${\sf d}(C_n \rtimes_s C_2) = n$ (see Lemma 6 of [Zhuang, Gao; Europ. J. Combin. 26 (2005), 1053-1059]). In this paper, we determine all sequences over $C_n \rtimes_s C_2$ of length $n$ which are product-one free. It completes the classification of all product-one free sequences over every group of the form $C_n \rtimes_s C_2$, including the quasidihedral groups and the modular maximal-cyclic groups.

math.NT↗

arXiv2021

The main zero-sum constants over $D_{2n} \times C_2$

Let $C_2$ be the cyclic group of order $2$ and $D_{2n}$ be the dihedral group of order $2n$, where $n$ is even. In this paper, we provide the exact values of some zero-sum constants over $D_{2n} \times C_2$, namely small Davenport constant, Gao constant, $η$-constant and Erd\H os-Ginzburg-Ziv constant. As a consequence, we prove the Gao's and Zhuang-Gao's Conjectures for this group. These are the first concrete results on zero-sum problems for a family of non-abelian groups of rank greater than $2$.

math.NT↗

arXiv2021

Generators of finite fields with prescribed traces

This paper explores the existence and distribution of primitive elements in finite field extensions with prescribed traces in several intermediate field extensions. Our main result provides an inequality-like condition to ensure the existence of such elements. We then derive concrete existence results for a special class of intermediate extensions.

math.NT↗

arXiv2020

Permutations from an arithmetic setting

Let $m, n$ be positive integers such that $m>1$ divides $n$. In this paper, we introduce a special class of piecewise-affine permutations of the finite set $[1, n]:=\{1, \ldots, n\}$ with the property that the reduction $\pmod m$ of $m$ consecutive elements in any of its cycles is, up to a cyclic shift, a fixed permutation of $[1, m]$. Our main result provides the cycle decomposition of such permutations. We further show that such permutations give rise to permutations of finite fields. In particular, we explicitly obtain classes of permutation polynomials of finite fields whose cycle decomposition and its inverse are explicitly given.

math.NT↗

arXiv2018

The subsums of zero-sum free sequences in finite cyclic groups

Let $\mathbb Z_n$ be the cyclic group of order $n \ge 3$ additively written. S. Savchev \& F. Chen (2007) proved that for each zero-sum free sequence $S = a_1 \bullet \dots \bullet a_t$ over $\mathbb Z_n$ of length $t > n/2$, there is an integer $g$ coprime to $n$ such that, if $\overline{r}$ denotes the least positive integer in the congruence class $r$ modulo $n$, then $\sum_{i=1}^t \overline{ga_i} < n$. Under the same hypothesis, in this paper we show that $$\left\{ \sum_{i \in Λ} \overline{ga_i} \;\; \Bigg| \;\; Λ\subset \{1,2,\dots,t\} \right\} = \left\{ 1, 2, \dots, \sum_{i=1}^t \overline{ga_i}\right\}.$$ It simplifies many calculations on inverse zero-sum problems.

math.NT↗

arXiv2017

Extremal product-one free sequences in Dihedral and Dicyclic groups

Let $G$ be a finite group, written multiplicatively. The Davenport constant of $G$ is the smallest positive integer $D(G)$ such that every sequence of $G$ with $D(G)$ elements has a non-empty subsequence with product $1$. Let $D_{2n}$ be the Dihedral Group of order $2n$ and $Q_{4n}$ be the Dicyclic Group of order $4n$. J. J. Zhuang and W. Gao (European J. Combin. 26 (2005), 1053-1059) showed that $D(D_{2n}) = n+1$ and J. Bass (J. Number Theory 126 (2007), 217-236) showed that $D(Q_{4n}) = 2n+1$. In this paper, we give explicit characterizations of all sequences $S$ of $G$ such that $|S| = D(G) - 1$ and $S$ is free of subsequences whose product is $1$, where $G$ is equal to $D_{2n}$ or $Q_{4n}$ for some $n$.

math.NT↗

arXiv2016

Extremal product-one free sequences in $C_q \rtimes_s C_m$

Let $G$ be a finite group, written multiplicatively. The Davenport constant of $G$ is the smallest positive integer $d$ such that every sequence of $G$ with $d$ elements has a non-empty subsequence with product $1$. Let $C_n \simeq \mathbb Z_n$ be the cyclic group of order $n$. Bass (2007) showed that the Davenport constant of the metacyclic group $C_q \rtimes_s C_m$, where $q$ is a prime number and $\text{ord}_q(s) = m \ge 2$, is $m+q-1$. In this paper, we explicit the form of all sequences $S$ of $C_q \rtimes_s C_m$, with $q+m-2$ elements, that are free of product-$1$ subsequences.

math.NT↗