DEMAT - Departamento de Matemática
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Navegando DEMAT - Departamento de Matemática por Autor "Avelar, Danilo Vilela"
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Item A note on Bass’ conjecture.(2023) Avelar, Danilo Vilela; Brochero Martinez, Fabio Enrique; Ribas, SávioFor a finite group G, we denote by d(G) and by E(G), respectively, the small Davenport constant and the Gao constant of G. Let Cn be the cyclic group of order n and let Gm,n,s = Cn s Cm be a metacyclic group. In [2, Conjecture 17], Bass conjectured that d(Gm,n,s) = m + n − 2 and E(Gm,n,s) = mn + m + n − 2 provided ordn(s) = m. In this paper, we show that the assumption ordn(s) = m is essential and cannot be removed. Moreover, if we suppose that Bass’ conjecture holds for Gm,n,s and the mn-product-one free sequences of maximal length are well behaved, then Bass conjecture also holds for G2m,2n,r, where r2 ≡ s (mod n).Item On the direct and inverse zero-sum problems over Cn ⋊s C2.(2022) Avelar, Danilo Vilela; Brochero Martinez, Fabio Enrique; Ribas, SávioLet Cn be the cyclic group of order n. In this paper, we provide the exact values of some zero-sum constants over Cn ⋊s C2 where s 6≡ ±1 (mod 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 Cn ⋊s C2 of maximum length.