On a Generalization of the Cauchy-Davenport Theorem

A generalization of the Cauchy-Davenport Theorem to arbitrary finite groups was suggested by Károlyi and proved independently by Károlyi and Wheeler. Here we give 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| min{p(G), |A| + |B| 1} where p(G) denotes the smallest order of a nontrivial finite subgroup of G, or 1 if no such subgroups exist.