Combinatorics, Probability and Computing

Paper

Quasirandom Groups

W. T. GOWERSa1

a1 Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK (e-mail: w.t.gowers@dpmms.cam.ac.uk)

Abstract

Babai and Sós have asked whether there exists a constant c > 0 such that every finite group G has a product-free subset of size at least c|G|: that is, a subset X that does not contain three elements x, y and z with xy = z. In this paper we show that the answer is no. Moreover, we give a simple sufficient condition for a group not to have any large product-free subset.

(Received July 05 2006)

(Revised October 27 2007)