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arXiv · 2408.06278

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

Joseph Atalaye·Liam Baker·Sophie Marques
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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 \( 0 \leq f \leq e \).

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Joseph Atalaye, Liam Baker, Sophie Marques. 2024-08-12. On the automorphism group of the monoid of the integers modulo a prime power. https://arxiv.org/abs/2408.06278

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