平移稳定群的标准基与 wreath 积中的子群成员问题
Standard bases for shift-stable groups and Subgroup Membership in wreath products
- Magdalen College, University of Oxford（牛津大学莫德林学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究为有限群受限直积的平移稳定子群建立标准基概念，受 Buchberger 算法启发构造算法解决相关问题，进而证明有限群与自然数 $n$ 对应的 wreath 积中子群成员问题可判定。
AI中文摘要：
我们为受限直积 $G^{(\mathbb{N}^n)}$ 的子群建立了标准基的概念，其中 $G$ 为任意有限群，该子群在 $\mathbb{N}^n$ 的平移作用下保持稳定。我们构造了一种算法，可计算此类子群的标准基，并利用它们解决多项算法问题，包括成员问题、饱和问题与变量消去问题。我们的方法受 Buchberger 算法及多项式环理想的 Gröbner 基理论启发。基于上述标准基与算法问题的解决方案，我们证明：对于有限群 $G$ 及自然数 $n$，wreath 积 $G \wr \mathbb{Z}^n$ 中的子群成员问题是可判定的。
英文摘要：
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 $n \in \mathbb{N}$.
