Permutation groups generated by binomials

Let G(q) be the group of permutations of the finite field F_q which is generated by those permutations that can be written as c-->ac^m+bc^n with 0<m<n<q and a,b in F_q with ab nonzero. We show that there are infinitely many q for which G(q) is the group of all permutations of F_q which fix 0. This resolves a conjecture of Vasilyev and Rybalkin.