We investigate the power of counting in Group Isomorphism. We first leverage the count-free variant of the Weisfeiler–Leman Version I algorithm for groups [J. Brachter and P. Schweitzer, On the Weisfeiler–Leman dimension of finite groups, in 35th Annual ACM/IEEE Symp. Logic in Computer Science, eds. H. Hermanns, L. Zhang, N. Kobayashi and D. Miller, Saarbrucken, Germany, July 8–11, 2020 (ACM, 2020), pp. 287–300, doi:10.1145/3373718.3394786] in tandem with bounded non-determinism and limited counting to improve the parallel complexity of isomorphism testing for several families of groups. These families include:

• Direct products of non-Abelian simple groups.

• Coprime extensions, where the normal Hall subgroup is Abelian and the complement is an $O(1)$-generated solvable group with solvability class poly log log $n$. This notably includes instances where the complement is an $O(1)$-generated nilpotent group. This problem was previously known to be in $\mathsf{P}$ [Y. Qiao, J. M. N. Sarma and B. Tang, On isomorphism testing of groups with normal Hall subgroups, in Proc. 28th Symp. Theoretical Aspects of Computer Science, Dagstuhl Castle, Leibniz Center for Informatics, 2011), pp. 567–578, doi:10.4230/LIPIcs. STACS.2011.567], and the complexity was recently improved to $\mathsf{L}$ [J. A. Grochow and M. Levet, On the parallel complexity of group isomorphism via Weisfeiler–Leman, in 24th Int. Symp. Fundamentals of Computation Theory, eds. H. Fernau and K. Jansen, Lecture Notes in Computer Science, Vol. 14292, September 18–21, 2023, Trier, Germany (Springer, 2023), pp. 234–247].

• Graphical groups of class $2$ and exponent $p>2$ [A. H. Mekler, Stability of nilpotent groups of class 2 and prime exponent, J. Symb. Logic46(4) (1981) 781–788] arising from the CFI and twisted CFI graphs [J. -Y. Cai, M. Fürer and N. Immerman, An optimal lower bound on the number of variables for graph identification, Combinatorica12(4) (1992) 389–410], respectively. In particular, our work improves upon previous results of Brachter and Schweitzer [On the Weisfeiler–Leman dimension of finite groups, in 35th Annual ACM/IEEE Symp. Logic in Computer Science, eds. H. Hermanns, L. Zhang, N. Kobayashi and D. Miller, Saarbrucken, Germany, July 8–11, 2020 (ACM, 2020), pp. 287–300, doi:10.1145/3373718.3394786].

Notably, each of these families was previously known to be identified by the counting variant of the more powerful Weisfeiler–Leman Version II algorithm. We finally show that the $q$-ary count-free pebble game is unable to even distinguish Abelian groups. This extends the result of Grochow and Levet (ibid), who established the result in the case of $q=1$. The general theme is that some counting appears necessary to place Group Isomorphism into $\mathsf{P}$.

我们研究了群同构中计数的威力。我们首先利用 Weisfeiler-Leman 版本 I 算法的无计数变体来进行组 [J. Brachter 和 P. Schweitzer，论有限群的 Weisfeiler-Leman 维数，第 35 届年度 ACM/IEEE Symp。计算机科学中的逻辑，编辑。 H. Hermanns、L. 张、N. Kobayashi 和 D. Miller，萨尔布吕肯，德国，2020 年 7 月 8-11 日（ACM，2020），第 287-300 页，doi:10.1145/3373718.3394786] 与有界非串联确定性和有限计数，以提高多个组族同构测试的并行复杂性。这些家庭包括：

• 非阿贝尔单群的直积。

• 互质扩展，其中正规霍尔子群是阿贝尔子群，补数是$氧（1）$- 生成可解群，其可解性等级为poly log log$n$。这尤其包括补体是$氧（1）$-生成幂零群。此前已知此问题存在于$\mathsf{磷}$[Y。 Qiao、JMN Sarma 和 B. Tang，关于具有正态霍尔子群的群的同构检验，Proc.第 28 次症状。计算机科学的理论方面， Dagstuhl Castle，莱布尼茨信息学中心，2011 年），第 567-578 页，doi：10.4230/LIPIcs。 STACS.2011.567]，并且复杂度最近改进为$\mathsf{L}$[JA Grochow 和 M. Levet，通过 Weisfeiler-Leman 论群同构的并行复杂性，第 24 期 Int。症状。计算理论基础，编辑。 H. Fernau 和 K. Jansen，计算机科学讲义，卷。 14292，2023 年 9 月 18-21 日，德国特里尔（Springer，2023），第 234-247 页]。

• 班级图形组$2$和指数$p>2$[AH Mekler，2 类幂零群和素指数的稳定性，J. Symb。 Logic 46 (4) (1981) 781–788] 由 CFI 和扭曲的 CFI 图产生 [J。 -Y。 Cai, M. Fürer 和 N. Immerman，图识别变量数量的最佳下界，Combinatorica 12 (4) (1992) 389–410]，分别。特别是，我们的工作改进了 Brachter 和 Schweitzer 先前的结果 [关于有限群的 Weisfeiler-Leman 维数，在第 35 届 ACM/IEEE Symp 年度会议上。计算机科学中的逻辑，编辑。 H. Hermanns、L. 张、N. Kobayashi 和 D. Miller，德国萨尔布吕肯，2020 年 7 月 8-11 日（ACM，2020），第 287-300 页，doi:10.1145/3373718.3394786]。

值得注意的是，之前已知这些家族中的每一个都是通过更强大的 Weisfeiler-Leman Version II 算法的计数变体来识别的。我们最终证明了$q$-无计数卵石游戏甚至无法区分阿贝尔群。这扩展了 Grochow 和 Levet（同上）的结果，他们在以下情况下得出了结果：$q=1$。总的主题是，一些计数似乎有必要将群同构放入$\mathsf{磷}$。