Year: 2026
Type: preprint
Abstract: We develop a notion of standard bases for subgroups of the restricted direct product $G^{(\mathbb{N}^n)}$ that are stable under translation by $\mathbb{N}^n$, where $G$ is an arbitrary finite group. We construct an algorithm that computes standard bases for such subgroups and use them to solve several algorithmic problems, including membership, saturation, and variable elimination. Our approach is inspired by Buchberger's algorithm and the theory of Gröbner bases for ideals in polynomial rings. Building on the standard bases and our solutions to the algorithmic problems above, we prove that Subgroup Membership is decidable in wreath products $G \wr \mathbb{Z}^n$ for finite $G$ and $n \in \mathbb{N}$.
Source: arXiv (Cornell University)
Author: Dong, Ruiwen
Institution: University of Oxford
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204234186