Effective results on the Waring problem for finite simple groups

Abstract

Let $G$ be a finite quasisimple group of Lie type. We show that there are regular semisimple elements $x, y\in G$, $x$ of prime order, and $|y|$ is divisible by at most two primes, such that $x^G\cdot y^G\supseteq G\setminus Z(G)$. In fact in all but four cases, $y$ can be chosen to be of square-free order. Using this result, we prove an effective version of a previous result of Larsen, Shalev, and Tiep by showing that, given any integer $m\geq 1$, if the order of a finite simple group $S$ is at least $m^{8m^2}$, then every element in $S$ is a product of two $m$th powers. Furthermore, the verbal width of $x^m$ on any finite simple group $S$ is at most $80m\sqrt{2\log_2 m}+56$. We also show that, given any two non-trivial words $w_1$, $w_2$, if $G$ is a finite quasisimple group of large enough order, then $w_1(G)w_2(G) \supseteq G\setminus Z(G)$.

Locations

• American Journal of Mathematics - View
• arXiv (Cornell University) - View - PDF