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，我们利用无计数的Weisfeiler–Leman版本I算法（Brachter & Schweitzer, LICS 2020）的群论变体，结合有限非确定性和有限计数，改进了若干群族同构检测的并行复杂度。这些群族包括：- 非阿贝尔单群的直积。- 互质扩张，其中正规霍尔子群为阿贝尔群，补群为$O(1)$生成的可解群，且可解类为$\text{poly} \log \log n$。这尤其包含补群为$O(1)$生成的幂零群的情况。此前该问题已知属于$\textsf{P}$（Qiao, Sarma, & Tang, STACS 2011），最近复杂度被改进为$\textsf{L}$（Grochow & Levet, FCT 2023）。- 类$2$且指数$p > 2$的图群（Mekler, J. Symb. Log., 1981），这些图群分别由CFI图和扭曲CFI图（Cai, F\"urer, & Immerman, Combinatorica 1992）构造。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，$q$元无计数石子游戏甚至无法区分阿贝尔群。这推广了Grochow & Levet（同上）在$q=1$情况下的结果。总体主题是：将群同构归入$\textsf{P}$似乎需要某种计数机制。