A generalization of the diameter bound of Liebeck and Shalev for finite simple groups

Abstract

Let $G$ be a non-abelian finite simple group. A famous result of Liebeck and Shalev is that there is an absolute constant $c$ such that whenever $S$ is a non-trivial normal subset in $G$ then $S^k=G$ for any integer $k$ at least $c\cdot (\log |G|/\log |S|)$. This result is generalized by showing that there exists an absolute constant $c$ such that whenever $S_1,\dots ,S_k$ are normal subsets in $G$ with ${\prod }_{i=1}^k|S_i|\ge {|G|}^c$ then $S_1\cdots S_k=G$.