A Representation of Permutations with Full Cycle

Abstract

For q > 2, Carlitz proved that the group of permutation polynomials (PPs) over F_q is generated by linear polynomials and x^{q-2}. Based on this result, this note points out a simple method for representing all PPs with full cycle over the prime field F_p, where p is an odd prime. We use the isomorphism between the symmetric group S_p of p elements and the group of PPs over F_p, and the well-known fact that permutations in S_p have the same cycle structure if and only if they are conjugate.