A Representation of Permutations with Full Cycle

الأصل · EN

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