How abelian is a finite group?

Abstract

The first paper with the above title was written by Erdős and Straus. Here we solve one of the problems considered there by proving that every group of order n contains an abelian subgroup of order at least 2ε√logn for some ε>0. This result is essentially best possible.We also give a quick survey of recent developments in related areas of group theory which were greatly stimulated by questions of Erdős.