21.110 (2026)

Let $S$ be a nonabelian finite simple group, and $x$ a nonidentity automorphism of $S$. Let $\alpha(x)$ be the smallest number of conjugates of $x$ in $G = \langle x, \text{Inn}\,S \rangle$ that generate $G$. The values of $\alpha(x)$ had been studied in (R. Guralnick, J. Saxl, J. Algebra, 268, no. 2 (2003), 519–571).
$\qquad$ a) (R. Guralnick, J. Saxl). Conjecture: If $S$ is an exceptional group of Lie type, then $\alpha(x) \leqslant 5$ for every nonidentity automorphism $x$ of $S$.
$\qquad$ b) For each exceptional group $S$ of Lie type, find the largest value of $\alpha(x)$.