PreprintCC BY 4.0arXiv
Standard bases for shift-stable groups and Subgroup Membership in wreath products
Abstract
We develop a notion of standard bases for subgroups of the restricted direct product G(ℕn) that are stable under translation by ℕn, 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 ℤn for finite G and n ∈ ℕ.
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BibTeX
@misc{dong2026standard,
title = {{Standard bases for shift-stable groups and Subgroup Membership in wreath products}},
author = {Ruiwen Dong},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.24620},
eprint = {2608.24620},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.24620},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Standard bases for shift-stable groups and Subgroup Membership in wreath products
AU - Ruiwen Dong
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.24620
UR - https://arxiv.org/abs/2608.24620
AB - We develop a notion of standard bases for subgroups of the restricted direct product G^(ℕ^n) that are stable under translation by 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 ℤ^n for finite G and n ∈ ℕ.
ER -