Some results on minimal sumset sizes in finite non-Abelian groups. (Q877935)

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 \(\mu_G(r,s)\geq\kappa_G(r,s)\).