= Diameter bound for a non-abelian finite simple group
For every $\varepsilon>0$ there is $\lambda(\varepsilon)$ such that every non-abelian finite <simple group> $G$ and every finite symmetric generating set $S$ containing the identity satisfy
$$
\operatorname{diam}_S(G)\leq\max\{|G|^\varepsilon,\lambda(\varepsilon)\}.
$$
The 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.
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