On a generalization of the Cauchy-Davenport theorem (Q2830347)

The Cauchy-Davenport Theorem is a fundamental result in combinatorial number theory, which has been generalized in different ways. A generalization of the Cauchy-Davenport Theorem to arbitrary finite groups was suggested by Károlyi and proved independently by \textit{G. Károlyi} [Enseign. Math. (2) 51, No. 3--4, 239--254 (2005; Zbl 1111.20026)] and \textit{J. P. Wheeler} [``The Cauchy-Davenport theorem for finite groups'', Preprint, \url{arXiv:1202.1816}]. In this paper, the author gives a short proof of the following small extension of this result (which also applies to infinite groups): If $A$, $B$ are finite nonempty subsets of a (multiplicatively written) group $G$ then $|AB| \geq \min\{p(G), |A|+|B|-1\}$ where $p(G)$ denotes the smallest order of a nontrivial finite subgroup of $G$, or $\infty$ if no such subgroups exist.