Subgroup and Submonoid Membership in the lampshuffler of $\mathbb{Z}$

2026-10-05

AI Summary

This study addresses the membership problem for subgroups and subsemigroups within the integer lamplighter group FSym(Z)⋊Z. Methodologically, by integrating group theory, semidirect product structural analysis, and wreath product theory, the work employs computational complexity reductions to transform the target problems into known decidable problems over wreath products. The core contribution lies in establishing, for the first time, the algorithmic decidability of subgroup and subsemigroup membership within this specific infinite permutation group structure. Consequently, this research provides a novel decision framework and methodological foundation for the computational theory of infinite groups.

Abstract

The lampshuffler group of $\mathbb{Z}$ is the semidirect product $\operatorname{FSym}(\mathbb{Z}) \rtimes \mathbb{Z}$, which consists of all permutations of $\mathbb{Z}$ that act as a translation outside a finite set. This infinite permutation group naturally contains as subgroups the wreath products $H \wr \mathbb{Z}$ for every finite group $H$. We prove that the Subgroup Membership Problem, and more generally, the Submonoid Membership Problem, are decidable in $\operatorname{FSym}(\mathbb{Z}) \rtimes \mathbb{Z}$. Our proof reduces Subgroup and Submonoid Membership in $\operatorname{FSym}(\mathbb{Z}) \rtimes \mathbb{Z}$ to Subgroup Membership in the wreath products $H \wr \mathbb{Z}$, which was shown to be decidable by Lohrey, Steinberg and Zetzsche (2015).

Problem

• Subgroup Membership Problem
• Submonoid Membership Problem
• lampshuffler group
• wreath product
• decidability

Innovation

• Subgroup Membership Problem
• Submonoid Membership Problem
• lampshuffler group
• wreath products
• decidability