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

TL;DR

This paper characterizes the automorphism groups of the unit group and the monoid of integers modulo prime powers, revealing their detailed algebraic structures for large exponents.

Contribution

It provides a detailed description of the automorphism groups of these algebraic structures, including explicit group decompositions for prime power moduli.

Findings

Automorphism group of ext{U}_{2^e} is a direct product involving dihedral and cyclic groups.

Automorphism group of ext{Z}/p^e ext{Z} is a semidirect product of units and automorphisms.

Results hold for all e ≥ 5, with explicit group structures identified.

Abstract

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