A note on the multiplicative structure of Z/pᵉZ

This paper determines the structure of the automorphism group of the monoid \((Z/p^e Z, ·)\) and the group of units \((Up^e, ·)\). We prove that the automorphism group of \((Z/p^e Z, ·)\) is isomorphic to a canonical semidirect product of \(U_{pᵉ⁻¹}\) and \((Up^e, ·)\), except when \(p^e = 8\), where it is isomorphic to the Klein group. Additionally, for \(e ≥ 5\), the automorphism group of \((U2^e, ·)\) is the direct product of \(Z/2Z\) with the central product of a dihedral group of order 8 and the cyclic group \(Z/2ᵉ⁻³Z\). Keywords. Multiplicative automorphisms, Monoid, Z/pᵉZ, Group theory, Algebraic structures, Semi-direct product, Unit group, Dihedral group, Central product, Ring theory, Multiplicative automorphisms.