On Finite Simple Groups with the Set of Element Orders as in a Frobenius Group or a Double Frobenius Group

It is proved that a finite simple group with the set of element orders as in a Frobenius group (a double Frobenius group, respectively) is isomorphic to $L_3(3)$ or $U3_(3)$ (to $U_3(3)$ or $S_4(3)$, respectively).