Permutations in a finite field
L. Carlitz · Proceedings of the American Mathematical Society · 1953
(1) ax + f, xQ-2 (at: EGF(q), a # 0). For q = 5, this was proved to be true by Betti; for q =7 the corresponding result was verified by Dickson [1, p. 119]. In this note we show very simply that this result holds for all q. Since the totality of permutation polynomials evidently furnishes a representation of the symmetric group on q letters, it will suffice to show that every transposition (Ga) can be generated by means of the special polynomials (1); here a denotes a fixed nonzero number CGF(q). We consider the following polynomial