# Diameters of finite simple groups: sharp bounds and applications

### Abstract

Let $G$ be a finite simple group and let $S$ be a normal subset of $G$. We determine the diameter of the Cayley graph $\Gamma(G,S)$ associated with $G$ and $S$, up to a multiplicative constant. Many applications follow. For example, we deduce that there is a constant $c$ such that every element of $G$ is a product of $c$ involutions (and we generalize this to elements of arbitrary order). We also show that for any word $w=w(x_1,\ldots,x_d)$, there is a constant $c = c(w)$ such that for any simple group $G$ on which $w$ does not vanish, every element of $G$ is a product of $c$ values of $w$. From this we deduce that every verbal subgroup of a semisimple profinite group is closed. Other applications concern covering numbers, expanders, and random walks on finite simple groups.

DOI

DOI: https://doi.org/10.2307/3062101

## Authors

Martin W. Liebeck

Aner Shalev