17.73 (2010)

Let $G$ be a finite simple group of Lie type defined over a field of characteristic $p$, and $V$ an absolutely irreducible $G$-module over a field of the same characteristic. Is it true that in the following cases the split extension of $V$ by $G$ must contain an element whose order is distinct from the order of any element of $G$?
$\qquad$ a) $G = U_4(q)$;
$\qquad$ b) $G = S_{2n}(q)$, $n \geqslant 3$;
$\qquad$ c) $G = O_{2n+1}(q)$, $n \geqslant 3$;
$\qquad$ d) $G = O^+_{2n}(q)$, $n \geqslant 4$;
$\qquad$ e) $G = O^-_{2n}(q)$, $n \geqslant 4$;
$\qquad$ f) $G = {}^3D_4(q)$, $q \neq 2$;
$\qquad$ g) $G = E_6(q)$;
$\qquad$ h) $G = {}^2E_6(q)$;
$\qquad$ i) $G = E_7(q)$;
$\qquad$ j) $G = G_2(2^m)$.