On the automorphism group of the monoid of the integers modulo a prime power

This paper determines the structure of the automorphism group of the unit group $(U_{p^e},\cdot )$ and the monoid $(\mathbb{Z}\mathrm{/}{p}^e\mathbb{Z},\cdot )$. For $e\ge 5$, we establish that the automorphism group $\text{Aut}(U_{2^e},\cdot )$ is the direct product of $\mathbb{Z}\mathrm{/}2\mathbb{Z}$ with the central product of a dihedral group of order 8 and the cyclic group $\mathbb{Z}\mathrm{/}{2}^{e-3}\mathbb{Z}$. Moreover, we show that the automorphism group $\text{Aut}(\mathbb{Z}\mathrm{/}{p}^e\mathbb{Z},\cdot )$ is isomorphic to a canonical semidirect product of $U_{p^{e-1}}$ and the subgroup of $\text{Aut}(U_{p^e},\cdot )$ consisting of automorphisms that induce an automorphism of $(U_{p^f},\cdot )$ for any integer $f$ such that $0\le f\le e$.