On products of sets in groups

Abstract

Let G a group, a finite subset of G containing 1,A a non empty finite subset of G. Olson [6] proved that either$AB = A\left\langle B \right\rangle or \left| {AB} \right| \geqslant \left| A \right| + \left\lfloor {\raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} (\left| B \right| + 1)} \right\rfloor$. Using the connectivity of Cayley graphs, we have another proof of this result and show moreover that either$AB = A\left\langle B \right\rangle or \left| {AB} \right| \geqslant \left| A \right| + \left\lfloor {\frac{2}{3}(\left| B \right| + 1)} \right\rfloor$ if B = B ⁻¹ or B ∩ B ⁻¹ = {1}..