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

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