Quasirandom groups

TL;DR

This paper proves that not all finite groups contain large product-free subsets and provides a simple condition to identify groups lacking such subsets, answering a question posed by Babai and Sós.

Contribution

It demonstrates that the existence of large product-free subsets is not universal in finite groups and offers a criterion to determine groups without large such subsets.

Findings

Not all finite groups have large product-free subsets.

Provides a simple sufficient condition for groups lacking large product-free subsets.

Answers a question posed by Babai and Sós.

Abstract

Babai and S\'os 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.