# Product decompositions of quasirandom groups and a Jordan type theorem

### Nikolay Nikolov

University of Oxford, UK### László Pyber

Hungarian Academy of Sciences, Budapest, Hungary

## Abstract

We first note that a result of Gowers on product-free sets in groups has an unexpected consequence: If $k$ is the minimal degree of a representation of the finite group~$G$, then for every subset $B$ of $G$ with $|B| > |G| / k^{\frac13}$ we have $B^3 = G$. We use this to obtain improved versions of recent deep theorems of Helfgott and of Shalev concerning product decompositions of finite simple groups, with much simpler proofs. On the other hand, we prove a version of Jordan's theorem which implies that if $k \geq 2$, then $G$ has a proper subgroup of index at most $c_0 k^2$ for some constant $c_0$, hence a product-free subset of size at least $|G| / ck$. This answers a question of Gowers.

## Cite this article

Nikolay Nikolov, László Pyber, Product decompositions of quasirandom groups and a Jordan type theorem. J. Eur. Math. Soc. 13 (2011), no. 4, pp. 1063–1077

DOI 10.4171/JEMS/275