Growth in ${\mathrm{SL}}_3(Z/pZ)$

Abstract

Let $G={\mathrm{SL}}_3$(ℤ/$p$ℤ), $p$ a prime. Let $A$ be a set of generators of $G$. Then $A$ grows under the group operation.

To be precise: denote by $|S|$ the number of elements of a finite set $S$. Assume $|A|<|G{|}^{1-\epsilon }$ for some $\epsilon >0$. Then $|A\cdot A\cdot A|>|A{|}^{1+\delta }$, where $\delta >0$ depends only on $\epsilon$.

We will also study subsets $A\subset G$ that do not generate $G$. Other results on growth and generation follow.