On the conjugacy class exponent of the finite simple groups

The generalized order $e_G(g)$ of an element $g$ of a group $G$ is the smallest positive integer $k$ such that there exist $x_1,\ldots,x_k \in G$ such that $g^{x_1} \ldots g^{x_k}=1$, where $g^x=x^{-1}gx$. Let $e(G) = \max \{e_G(g)\ |\ g \in G\}$. We provide upper bounds for $e(G)$ for every finite simple group $G$. In particular, we show that $e(G)\leq 8$ unless $G\in\{\mbox{PSL}_n(q), \mbox{PSU}_n(q), E_6(q),{^2}E_6(q)\}$. For the latter groups $e(G)\leq n,3n+3,36,36$, respectively. In addition, we bound from above the generalized order of semisimple and unipotent elements of finite simple groups of Lie type.