Group algebras of finite abelian groups and their applications to combinatorial problems
type: Item
instance of: scholarly article
title: Group algebras of finite abelian groups and their applications to combinatorial problems
zbMATH Open document ID: 1200.05254
DOI: 10.1216/RMJ-2009-39-3-805
author: Weidong Gao
author: Alfred Geroldinger
author: Franz Halter-Koch
published in: Rocky Mountain Journal of Mathematics
publication date: 2009-06-18T00:00:00Z
review text: Group algebras have turned out to be powerful tools for a variety of questions from combinatorics and number theory. For example, in zero-sum problems, the Davenport constant of \(G\), denoted by \(D(G)\), is defined to be the smallest positive integer \(d\) such that every sequence of \(d\) elements in \(G\) contains a nonempty zero-sum subsequence. The problem of finding \(D(G)\) was proposed by H. Davenport in 1966, and he also pointed out that \(D(G)\) is connected to algebraic number theory in the following way. Let \(K\) be an algebraic number field and \(G\) be its class group. Then \(D(G)\) is the maximal number of prime ideals (counting multiplicity) that can occur in the decomposition of an irreducible integer in \(K\). It plays an important role in unique factorization theory in algebraic number theory. The study of \(D(G)\) has attracted a great deal of attention. And the exact value of \(D(G)\) has been determined only for a few classes of groups. Group rings were tried to investigate \(D(G)\). In this paper, the authors considered the following crucial problem which connected with many problems in additive number theory: under what conditions, the sequence \(S=g_1\cdot \ldots \cdot g_l\) over finite abelian group \(G\) satisfies \[ f=(X^{g_1}-a_1)\cdot \ldots \cdot(X^{g_l}-a_l)\neq 0\in R[G] \text{ for all }a_1, \ldots, a_l \in R\setminus \{0\} \] .
reviewed by: Jujuan Zhuang
Mathematics Subject Classification ID: 05E15
Mathematics Subject Classification ID: 20K01
Mathematics Subject Classification ID: 11B50
Mathematics Subject Classification ID: 05B15
zbMATH DE Number: 5566202
zbMATH Keywords: group algebras
zbMATH Keywords: finite abelian groups
zbMATH Keywords: zero-sum sequence
zbMATH Keywords: additive Latin squares
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5566202