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 (Brachter & Schweitzer, LICS 2020) in tandem with limited 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 $\text{poly} \log \log n$. This notably includes instances where the complement is an $O(1)$-generated nilpotent group. This problem was previously known to be in $\textsf{P}$ (Qiao, Sarma, & Tang, STACS 2011), and the complexity was recently improved to $\textsf{L}$ (Grochow & Levet, FCT 2023). - Graphical groups of class $2$ and exponent $p > 2$ (Mekler, J. Symb. Log., 1981) arising from the CFI and twisted CFI graphs (Cai, F\"urer, & Immerman, Combinatorica 1992) respectively. In particular, our work improves upon previous results of Brachter & Schweitzer (LICS 2020). We finally show that the $q$-ary count-free pebble game is unable to distinguish even Abelian groups. This extends the result of Grochow & 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 $\textsf{P}$.

翻译：我们研究了计数在群同构问题中的能力。首先，我们利用群的Weisfeiler—Leman第一版算法的无计数变体（Brachter & Schweitzer, LICS 2020），结合有限非确定性和有限计数，改进了几类群同构测试的并行复杂度。这些群类包括：- 非阿贝尔单群的直积。- 互质扩张，其中正规Hall子群为阿贝尔群，补群为$O(1)$-生成的可解群且可解类为$\text{poly} \log \log n$。这特别包括补群为$O(1)$-生成的幂零群的情形。该问题先前已知属于$\textsf{P}$（Qiao, Sarma, & Tang, STACS 2011），且复杂度近期被改进为$\textsf{L}$（Grochow & Levet, FCT 2023）。- 类$2$且指数$p > 2$的图形群（Mekler, J. Symb. Log., 1981），分别源于CFI图和扭曲CFI图（Cai, F\"urer, & Immerman, Combinatorica 1992）。特别地，我们的工作改进了Brachter & Schweitzer（LICS 2020）的先前结果。最后，我们证明$q$元无计数卵石游戏甚至无法区分阿贝尔群。这推广了Grochow & Levet（同前）在$q = 1$情形下的结果。总体主题是：将群同构问题归入$\textsf{P}$需要某种程度的计数。