The group isomorphism problem determines whether two groups, given by their Cayley tables, are isomorphic. For groups with order $n$, an algorithm with $n^{(\log n + O(1))}$ running time, attributed to Tarjan, was proposed in the 1970s [Mil78]. Despite the extensive study over the past decades, the current best group isomorphism algorithm has an $n^{(1 / 4 + o(1))\log n}$ running time [Ros13]. The isomorphism testing for $p$-groups of (nilpotent) class 2 and exponent $p$ has been identified as a major barrier to obtaining an $n^{o(\log n)}$ time algorithm for the group isomorphism problem. Although the $p$-groups of class 2 and exponent $p$ have much simpler algebraic structures than general groups, the best-known isomorphism testing algorithm for this group class also has an $n^{O(\log n)}$ running time. In this paper, we present an isomorphism testing algorithm for $p$-groups of class 2 and exponent $p$ with running time $n^{O((\log n)^{5/6})}$ for any prime $p > 2$. Our result is based on a novel reduction to the skew-symmetric matrix tuple isometry problem [IQ19]. To obtain the reduction, we develop several tools for matrix space analysis, including a matrix space individualization-refinement method and a characterization of the low rank matrix spaces.

翻译：群同构问题旨在判定两个以凯莱表给定的群是否同构。对于阶数为$n$的群，Tarjan于1970年代提出运行时间为$n^{(\log n + O(1))}$的算法[Mil78]。尽管过去数十年间研究广泛，当前最优的群同构算法运行时间仍为$n^{(1/4+o(1))\log n}$[Ros13]。（幂零）类2且指数为$p$的$p$-群的同构测试已被视为实现群同构问题$n^{o(\log n)}$时间算法的关键瓶颈。尽管类2指数$p$的$p$-群具有比一般群更简单的代数结构，该类群已知最优的同构测试算法仍需$n^{O(\log n)}$运行时间。本文提出针对类2指数$p$的$p$-群的同构测试算法，对于任意素数$p>2$，其运行时间为$n^{O((\log n)^{5/6})}$。该结果基于对斜对称矩阵元组等距问题[IQ19]的创新性归约。为建立该归约，我们开发了矩阵空间分析的若干工具，包括矩阵空间个性化-精化方法及低秩矩阵空间的刻画。