The automorphism group of reduced power monoids of finite abelian groups

TL;DR

This paper characterizes the automorphism group of the reduced power monoid of a finite abelian group, showing it is isomorphic to the automorphism group of the group itself, with a specific exception.

Contribution

It provides a complete description of the automorphism group of reduced power monoids for finite abelian groups, extending previous work and identifying a special case involving the Klein four-group.

Findings

Automorphism group of reduced power monoid is isomorphic to that of the original group.

The isomorphism holds generally except for the Klein four-group.

The result clarifies the structure of automorphisms in this algebraic context.

Abstract

Let $H$ be an additively written monoid and let $\mathcal{P}_{0}(H)$ denote the reduced power monoid of $H$, that is, the monoid consisting of all subsets of $H$ containing $0$ with set addition as operation. Following work of Tringali, Wen and Yan, we give a full description of the automorphism group of $\mathcal{P}_{0}(G)$, where $G$ is a finite abelian group. More precisely, we show that $\text{Aut}(\mathcal{P}_{0}(G))$ and $\text{Aut}(G)$ are isomorphic in a canonic way, except in the special case when $G$ is isomorphic to the Klein four-group.