Number of solutions in a box of a linear equation in an Abelian group

Maciej Zakarczemny

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Displaying similar documents to “Number of solutions in a box of a linear equation in an Abelian group”

Self-small products of abelian groups

Josef Dvořák, Jan Žemlička (2022)

Commentationes Mathematicae Universitatis Carolinae

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Let $A$ and $B$ be two abelian groups. The group $A$ is called $B$ -small if the covariant functor $Hom (A, -)$ commutes with all direct sums $B^{(κ)}$ and $A$ is self-small provided it is $A$ -small. The paper characterizes self-small products applying developed closure properties of the classes of relatively small groups. As a consequence, self-small products of finitely generated abelian groups are described.

On the derived length of units in group algebra

Dishari Chaudhuri, Anupam Saikia (2017)

Czechoslovak Mathematical Journal

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Let $G$ be a finite group $G$ , $K$ a field of characteristic $p \geq 17$ and let $U$ be the group of units in $K G$ . We show that if the derived length of $U$ does not exceed $4$ , then $G$ must be abelian.

On the structure of sequences with forbidden zero-sum subsequences

W. D. Gao, R. Thangadurai (2003)

Colloquium Mathematicae

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We study the structure of longest sequences in $ℤ ₙ^{d}$ which have no zero-sum subsequence of length n (or less). We prove, among other results, that for $n = 2^{a}$ and d arbitrary, or $n = 3^{a}$ and d = 3, every sequence of c(n,d)(n-1) elements in $ℤ ₙ^{d}$ which has no zero-sum subsequence of length n consists of c(n,d) distinct elements each appearing n-1 times, where $c (2^{a}, d) = 2^{d}$ and $c (3^{a}, 3) = 9$ .

Non-supersingular hyperelliptic jacobians

Yuri G. Zarhin (2004)

Bulletin de la Société Mathématique de France

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Let $K$ be a field of odd characteristic $p$ , let $f (x)$ be an irreducible separable polynomial of degree $n \geq 5$ with big Galois group (the symmetric group or the alternating group). Let $C$ be the hyperelliptic curve $y^{2} = f (x)$ and $J (C)$ its jacobian. We prove that $J (C)$ does not have nontrivial endomorphisms over an algebraic closure of $K$ if either $n \geq 7$ or $p \neq 3$ .

Some Additive $2 - (v, 5, λ)$ Designs

Andrea Caggegi (2015)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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Given a finite additive abelian group $G$ and an integer $k$ , with $3 \leq k \leq | G |$ , denote by $𝒟_{k} (G)$ the simple incidence structure whose point-set is $G$ and whose blocks are the $k$ -subsets $C = {c_{1}, c_{2}, \dots, c_{k}}$ of $G$ such that $c_{1} + c_{2} + \dots + c_{k} = 0$ . It is known (see [Caggegi, A., Di Bartolo, A., Falcone, G.: Boolean 2-designs and the embedding of a 2-design in a group arxiv 0806.3433v2, (2008), 1–8.]) that $𝒟_{k} (G)$ is a 2-design, if $G$ is an elementary abelian $p$ -group with $p$ a prime divisor of $k$ . From [Caggegi, A., Falcone, G., Pavone, M.: On the additivity...

How to construct a Hovey triple from two cotorsion pairs

James Gillespie (2015)

Fundamenta Mathematicae

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Let be an abelian category, or more generally a weakly idempotent complete exact category, and suppose we have two complete hereditary cotorsion pairs $(, \tilde{})$ and $(\tilde{},)$ in satisfying $\tilde{} \subseteq$ and $\cap \tilde{} = \tilde{} \cap$ . We show how to construct a (necessarily unique) abelian model structure on with (resp. $\tilde{}$ ) as the class of cofibrant (resp. trivially cofibrant) objects, and (resp. $\tilde{}$ ) as the class of fibrant (resp. trivially fibrant) objects.

Almost everywhere convergence of convolution powers on compact abelian groups

Jean-Pierre Conze, Michael Lin (2013)

Annales de l'I.H.P. Probabilités et statistiques

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It is well-known that a probability measure $μ$ on the circle $𝕋$ satisfies $∥ μ^{n} {* f - \int f d m ∥}_{p} \to 0$ for every $f \in L_{p}$ , every (some) $p \in [1, \infty)$ , if and only if $| \hat{μ} (n) | l t; 1$ for every non-zero $n \in ℤ$ ( $μ$ is strictly aperiodic). In this paper we study the a.e. convergence of $μ^{n} * f$ for every $f \in L_{p}$ whenever $p g t; 1$ . We prove a necessary and sufficient condition, in terms of the Fourier–Stieltjes coefficients of $μ$ , for the strong sweeping out property (existence of a Borel set $B$ with $lim sup μ^{n} * 1_{B} = 1$ a.e. and $lim inf μ^{n} * 1_{B} = 0$ a.e.). The results are extended to general compact Abelian groups...

On coverings of simple abelian varieties

Olivier Debarre (2006)

Bulletin de la Société Mathématique de France

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To any finite covering $f : Y \to X$ of degree $d$ between smooth complex projective manifolds, one associates a vector bundle $E_{f}$ of rank $d - 1$ on $X$ whose total space contains $Y$ . It is known that $E_{f}$ is ample when $X$ is a projective space ([Lazarsfeld 1980]), a Grassmannian ([Manivel 1997]), or a Lagrangian Grassmannian ([Kim Maniel 1999]). We show an analogous result when $X$ is a simple abelian variety and $f$ does not factor through any nontrivial isogeny $X^{'} \to X$ . This result is obtained by showing that $E_{f}$ is $M$ -regular...

Completely bounded lacunary sets for compact non-abelian groups

Kathryn Hare, Parasar Mohanty (2015)

Studia Mathematica

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In this paper, we introduce and study the notion of completely bounded $Λ_{p}$ sets ( $Λ_{p}^{c b}$ for short) for compact, non-abelian groups G. We characterize $Λ_{p}^{c b}$ sets in terms of completely bounded $L^{p} (G)$ multipliers. We prove that when G is an infinite product of special unitary groups of arbitrarily large dimension, there are sets consisting of representations of unbounded degree that are $Λ_{p}$ sets for all p < ∞, but are not $Λ_{p}^{c b}$ for any p ≥ 4. This is done by showing that the space of completely bounded $L^{p} (G)$ ...

A property which ensures that a finitely generated hyper-(Abelian-by-finite) group is finite-by-nilpotent

Fares Gherbi, Nadir Trabelsi (2024)

Czechoslovak Mathematical Journal

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Let $𝔐$ be the class of groups satisfying the minimal condition on normal subgroups and let $Ω$ be the class of groups of finite lower central depth, that is groups $G$ such that $γ_{i} (G) = γ_{i + 1} (G)$ for some positive integer $i$ . The main result states that if $G$ is a finitely generated hyper-(Abelian-by-finite) group such that for every $x \in G$ , there exists a normal subgroup $H_{x}$ of finite index in $G$ satisfying $〈 x, x^{h} 〉 \in 𝔐 Ω$ for every $h \in H_{x}$ , then $G$ is finite-by-nilpotent. As a consequence of this result, we prove that a finitely generated...

Polycyclic groups with automorphisms of order four

Tao Xu, Fang Zhou, Heguo Liu (2016)

Czechoslovak Mathematical Journal

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In this paper, we study the structure of polycyclic groups admitting an automorphism of order four on the basis of Neumann’s result, and prove that if $α$ is an automorphism of order four of a polycyclic group $G$ and the map $ϕ : G \to G$ defined by $g^{ϕ} = [g, α]$ is surjective, then $G$ contains a characteristic subgroup $H$ of finite index such that the second derived subgroup $H^{''}$ is included in the centre of $H$ and $C_{H} (α^{2})$ is abelian, both $C_{G} (α^{2})$ and $G / [G, α^{2}]$ are abelian-by-finite. These results extend recent and classical results in...