The automorphism group of reduced power monoids of finite abelian groups
Balint Rago
TL;DR
The paper characterizes the automorphism group of the reduced power monoid $\mathcal{P}_{0}(G)$ for a finite abelian group $G$. It introduces the pullback $g$ of an automorphism $f$ of $\mathcal{P}_{0}(G)$ and proves that $g$ is an automorphism of $G$, with $f$ equal to the augmentation of $g$; an induction on $|G|$ and a set of structural lemmas establish the canonical isomorphism $\mathrm{Aut}(\mathcal{P}_{0}(G)) \cong \mathrm{Aut}(G)$ for $G \not\cong C_2^2$. The Klein four group $C_2^2$ is the sole exception, where $\mathrm{Aut}(\mathcal{P}_{0}(G))$ is strictly larger (isomorphic to $S_3^2$) than $\mathrm{Aut}(G)$ (isomorphic to $S_3$). Overall, the work completes the description of automorphisms of reduced power monoids of finite abelian groups and clarifies when the natural embedding $\mathrm{Aut}(G) \hookrightarrow \mathrm{Aut}(\mathcal{P}_{0}(G))$ is surjective.
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.