\(t\)-sum generators of finite Abelian groups
type: Item
instance of: scholarly article
title: \(t\)-sum generators of finite Abelian groups
zbMATH Open document ID: 0774.20031
DOI: 10.1016/0012-365X(92)90320-F
author: Ron M. Roth
author: Abraham Lempel
published in: Discrete Mathematics
publication date: 1992-12-06T00:00:00Z
review text: Given a finite Abelian group \(A\) and an integer \(t\), \(1\leq t\leq| A|-1\), a subset \(S\) of \(A\) is called a \(t\)-sum generator of \(A\) if every element of \(A\) can be written as the sum of exactly \(t\) distinct elements of \(S\). The authors investigate the minimal integer \(M(t,A)\) such that every subset of \(A\) of size greater than \(M(t,A)\) is a \(t\)-sum generator of \(A\). (When \(A\) has no \(t\)-sum generator, \(M(t,A)=| A|.)\) For Abelian finite of even order, the authors get the following result: Let \(A\) be a finite Abelian group of even order \(\geq 12\). Then, for \(3\leq t\leq| A|/2-2\), \(M(t,A)=| A|/2\), except when \(A\in\{\mathbb{Z}^ h_ 2,\mathbb{Z}_ 4\times\mathbb{Z}_ 2^{h-1}\}\) and \(t\in\{3,| A|/2-2\}\), in which case \(M(t,A)=| A|/2+1\). For a group \(A\) of odd order, the problem of determining \(M(t,A)\) remains open. The case of odd-order groups seems to be much more complicated than that of even-order groups. The authors present some bounds on \(M(t,\mathbb{Z}^ h_ p)\), \(p\) an odd prime.
reviewed by: Yongcai Ren
Mathematics Subject Classification ID: 20K01
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 94B65
Mathematics Subject Classification ID: 11B75
zbMATH DE Number: 69891
zbMATH Keywords: \(t\)-sum generator
zbMATH Keywords: finite Abelian group
zbMATH Keywords: subset
zbMATH Keywords: odd-order groups
zbMATH Keywords: even- order groups
zbMATH Keywords: bounds
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 69891