Some results on minimal sumset sizes in finite non-Abelian groups.
type: Item
instance of: scholarly article
title: Some results on minimal sumset sizes in finite non-Abelian groups.
zbMATH Open document ID: 1130.20022
author: Shalom Eliahou
author: Michel Kervaire
published in: Journal of Number Theory
publication date: 2007-05-04T00:00:00Z
review text: Let \(G\) be a group, and let \(r\) and \(s\) be positive integers not exceeding \(|G|\). The authors [J. Algebra 287, No. 2, 449-457 (2005; Zbl 1095.11012)] proved that if \(G\) is Abelian then \[ \mu_G(r,s)=\min\{|AB|:A,B\subseteq G,\;|A|=r,\;|B|=s\} \] (with \(AB=\{ab:a\in A,\;b\in B\}\)) coincides with \[ \kappa_G(r,s)=\min_{d\in\mathcal H(G)}d(\lceil r/d\rceil+\lceil s/d\rceil-1), \] where \(\mathcal H(G)\) is the set of orders of all finite subgroups of \(G\). In the paper under review, the authors investigate \(\mu_G(r,s)\) for finite groups \(G\). They show that the equality \(\mu_G(r,s)=\kappa_G(r,s)\) remains valid if \(r+s\geq|G|\) or \(\kappa_G(r,s)<r/2+s\) or \(r\leq 3\). They also supply a counter-example to the equality in the case \(r=5\), and conjecture that \(\mu_G(r,s)\geq\kappa_G(r,s)\).
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 11B75
zbMATH DE Number: 5149382
zbMATH Keywords: sumset sizes
zbMATH Keywords: finite non-Abelian groups
zbMATH Keywords: Cauchy-Davenport theorem
reviewed by: Zhi-Wei Sun
MaRDI profile type: Publication
full work available at URL: j.jnt.2006.09.002
OpenAlex ID: W2017242648
cites work: On the Addition of Residue Classes
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DOI: 10.1016/J.JNT.2006.09.002
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description: scientific article; zbMATH DE number 5149382