# Cardinality of the Dickson permutation's group of polynomials on $\mathbb{Z}_n$

## TL;DR

This paper investigates the structure and size of the group of permutations induced by Dickson polynomials over integers modulo n, providing a method to compute its order.

## Contribution

It characterizes the group of Dickson polynomial permutations on Z_n and introduces an algorithm to determine its cardinality.

## Key findings

- Derived explicit descriptions of the permutation group G_n
- Developed an algorithm to compute |G_n| efficiently
- Solved systems of linear congruences related to G_n

## Abstract

Let $G_n$ be the group of permutations on $\mathbb{Z}_n$ that is induced by a Dickson polynomial, where $n$ is a positive integer. In this work, by solving special types of systems of linear congruences, we obtain $G_n$. In addition, we give an a algorithm that gets $|G_n|$.