The cycle index of the automorphism group of $\mathbb{Z}_n$

TL;DR

This paper computes the cycle index of the automorphism group of a5_n acting on residue classes modulo n, providing two methods for its derivation, which facilitates various enumerative and computational applications.

Contribution

It introduces two approaches to derive the cycle index of the automorphism group of a5_n, including an abstract method and a detailed composition-based method.

Findings

Derived the cycle index using an abstract approach.

Presented a detailed composition method for the cycle index.

Facilitated applications in enumeration and computation.

Abstract

We consider the group action of the automorphism group $\I_n=\aut(\Zz_n)$ on the set $\Zz_n$, that is the set of residue classes modulo $n$. Clearly, this group action provides a representation of $\I_n$ as a permutation group acting on $n$ points. One problem to be solved regarding this group action is to find its cycle index. Once it is found, there appears a vast class of related enumerative and computational problems with interesting applications. We provided the cycle index of specified group action in two ways. One of them is more abstract and hence compact, while another one is basically procedure of composing the cycle index from some \textit{building blocks}. However, those \textit{building blocks} are also well explained and finally presented in very detailed fashion.