A Representation of Permutations with Full Cycle

TL;DR

This paper presents a simple method to represent all permutation polynomials with a full cycle over prime fields, leveraging group isomorphisms and permutation cycle structures.

Contribution

It introduces a novel representation technique for full cycle permutation polynomials over prime fields based on group theory insights.

Findings

All permutation polynomials with full cycle over F_p can be represented using the proposed method.

The approach relies on the isomorphism between symmetric groups and permutation polynomials.

Cycle structure and conjugacy in symmetric groups are key to the representation.

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.