Abstract

A subset of an Abelian group is called sum-free if it contains no three elements x,y,z such that x+y=z. It is easy to prove that a cyclic group of size n contains a sum-free subset of size at least n/3, and this implies the same result for the product of a cyclic group with any other finite group—and hence for all finite Abelian groups. Babai and Sos asked whether a similar result was true for finite groups in general: is there a constant c>0 such that every group of order n contains a product-free subset of size at least cn? This talk will be about a property that many finite groups have, which is closely related to quasirandomness properties of graphs. It turns out that many natural families of groups, including all finite simple groups, have this property, and that no group with this property has a large product-free subset. Thus, the question of Babai and Sos has a negative answer for a typical “natural” finite non-Abelian group.