Diameter of a finite group with respect to a generating set (${\mathrm{diam}}_S(G)$)

For a finite group $G$ with a symmetric generating set $S$, its diameter isEquivalently, it is the diameter of the Cayley graph ${\mathrm{Cay}}_S(G)$.

A connected Cayley graph on a finite group $G$ has $|G|$ vertices, and a shortest path repeats no vertex. Therefore

For every $\epsilon >0$ there is $\lambda (\epsilon )$ such that every non-abelian finite simple group $G$ and every finite symmetric generating set $S$ containing the identity satisfyThe proof applies the one-scale virtual nilpotence theorem below the diameter and then uses simplicity to eliminate both the bounded-index subgroup and the finite normal subgroup.