超越奇特征：Frattini类2的2-群同构测试加速
Beyond odd characteristic: Faster isomorphism testing of 2-groups of Frattini class 2
- University of Colorado Boulder（科罗拉多大学博尔德分校）
- HUN-REN Institute for computer Science and Control（匈牙利科学院计算机与控制研究所）
- University of Technology Sydney（悉尼科技大学）
- University of Illinois Chicago（伊利诺伊大学芝加哥分校）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出首个对Frattini类2的2-群在$N^{O((\log N)^{1/2})}$时间内进行同构测试的算法，利用二次型等距问题和矩阵群算法，突破了$p=2$情况下的长期瓶颈。
AI中文摘要：
有限群同构问题询问两个阶为$N$的有限群是否同构。第一个算法归功于Tarjan（见Miller，STOC '78），运行时间为$N^{\log N + O(1)}$。尽管经过深入研究，当前已知最佳算法的运行时间为$N^{(1/4 + o(1))\log N}$（Rosenbaum，'13）。类2的$p$-群已被认为是加速群同构的主要瓶颈。最近的进展导致了对于$p$为奇数的类2 $p$-群的$N^{o(\log N)}$时间算法（Sun，STOC '23；Ivanyos--Mendoza--Qiao--Sun--Zhang，FOCS '24；Grochow--Qiao--Stange--Sun，STOC '25）。然而，$p=2$的情况，根据群枚举中一个著名的猜想，代表了类2 $p$-群的大多数，一直难以解决，直到现在几乎没有进展。在本文中，我们提出了一种算法，用于在$N^{O((\log N)^{1/2})}$时间内测试两个阶为$N$的Frattini类2的2-群是否同构。据我们所知，这是第一个针对一类2-群的$N^{o(\log N)}$时间同构算法，这类2-群在对数意义上构成了几乎所有2-群，即$\lim_{N \to \infty} \frac{\log(\text{阶} \leq N \text{的Frattini类2的2-群数量})}{\log(\text{阶} \leq N \text{的2-群数量})} = 1$。作为我们的主要工具，我们首次提出了在$\mathbb{F}_2$上的二次型空间/元组等距问题的非平凡算法。这些算法依赖于组合和代数思想的结合，包括Luks（FOCS '92）开发的有限矩阵群算法。据我们所知，这是首次使用矩阵群算法在$p$-群同构的最坏情况复杂度上取得进展。
英文摘要：
The finite group isomorphism problem asks whether two finite groups of order $N$ are isomorphic. The first algorithm, attributed to Tarjan (see Miller, STOC '78), runs in time $N^{\log N + O(1)}$. Despite intensive study, the current best known algorithm has a running time of $N^{(1 / 4 + o(1))\log N}$ (Rosenbaum, '13). $p$-groups of class $2$ have been recognized as the major bottleneck for faster group isomorphism. Recent progress has led to $N^{o(\log N)}$-time algorithms for $p$-groups of class $2$ where $p$ is odd (Sun, STOC '23; Ivanyos--Mendoza--Qiao--Sun--Zhang, FOCS '24; Grochow--Qiao--Stange--Sun, STOC '25). However, the case of $p=2$, which represents the majority of $p$-groups of class 2 assuming a well-known conjecture in group enumeration, remained elusive, with essentially no progress until now. In this paper, we present an algorithm for testing the isomorphism of two 2-groups of Frattini class 2 of order $N$ in time $N^{O((\log N)^{1/2})}$. To our knowledge, this is the first $N^{o(\log N)}$-time isomorphism algorithm for a class of $2$-groups that constitutes logarithmically almost all $2$-groups, in the sense that $\lim_{N \to \infty} \frac{\log(\text{\# 2-groups of Frattini class 2 and order } \leq N)}{\log(\text{\# 2-groups of order} \leq N)} = 1$. As our main tool, we present the first non-trivial algorithms for the quadratic form space/tuple isometry problems over $\mathbb{F}_2$. These algorithms rely on combinations of combinatorial and algebraic ideas, including finite matrix group algorithms developed by Luks (FOCS '92). As far as we know, this is the first time that matrix group algorithms are used to make progress on the worst-case complexity of $p$-group isomorphism.
