In this paper, we show that the constant-dimensional Weisfeiler-Leman algorithm for groups (Brachter & Schweitzer, LICS 2020) can be fruitfully used to improve parallel complexity upper bounds on isomorphism testing for several families of groups. In particular, we show: - Groups with an Abelian normal Hall subgroup whose complement is $O(1)$-generated are identified by constant-dimensional Weisfeiler-Leman using only a constant number of rounds. This places isomorphism testing for this family of groups into $\textsf{L}$; the previous upper bound for isomorphism testing was $\textsf{P}$ (Qiao, Sarma, & Tang, STACS 2011). - We use the individualize-and-refine paradigm to obtain a $\textsf{quasiSAC}^{1}$ isomorphism test for groups without Abelian normal subgroups, previously only known to be in $\textsf{P}$ (Babai, Codenotti, & Qiao, ICALP 2012). - We extend a result of Brachter & Schweitzer (arXiv, 2021) on direct products of groups to the parallel setting. Namely, we also show that Weisfeiler-Leman can identify direct products in parallel, provided it can identify each of the indecomposable direct factors in parallel. They previously showed the analogous result for $\textsf{P}$. We finally consider the count-free Weisfeiler-Leman algorithm, where we show that count-free WL is unable to even distinguish Abelian groups in polynomial-time. Nonetheless, we use count-free WL in tandem with bounded non-determinism and limited counting to obtain a new upper bound of $\beta_{1}\textsf{MAC}^{0}(\textsf{FOLL})$ for isomorphism testing of Abelian groups. This improves upon the previous $\textsf{TC}^{0}(\textsf{FOLL})$ upper bound due to Chattopadhyay, Tor\'an, & Wagner (ACM Trans. Comput. Theory, 2013).

翻译：本文表明，针对群的定维Weisfeiler-Leman算法（Brachter & Schweitzer, LICS 2020）可用于有效提升多类群族同构测试的并行复杂度上界。具体而言，我们证明了： - 具有阿贝尔正规Hall子群且其补群为$O(1)$生成的群族，可通过常轮数定维Weisfeiler-Leman算法识别。这将该群族的同构测试问题归入$\textsf{L}$复杂度类；此前已知最优上界为$\textsf{P}$（Qiao, Sarma, & Tang, STACS 2011）。 - 利用个体化-精炼范式，我们为无阿贝尔正规子群的群族建立了$\textsf{quasiSAC}^{1}$同构测试方法，此前该类群族仅被证明属于$\textsf{P}$（Babai, Codenotti, & Qiao, ICALP 2012）。 - 将Brachter & Schweitzer（arXiv, 2021）关于群直积的结果推广至并行场景：即证明若Weisfeiler-Leman算法能并行识别每个不可约直积因子，则其可并行识别直积群。此前他们仅得到$\textsf{P}$类中的类似结论。 最后，我们研究了无计数Weisfeiler-Leman算法，证明该算法甚至无法在多项式时间内区分阿贝尔群。尽管如此，通过将无计数WL与有界非确定性和有限计数技术结合，我们得到了阿贝尔群同构测试的新上界$\beta_{1}\textsf{MAC}^{0}(\textsf{FOLL})$，改进了Chattopadhyay, Torán, & Wagner（ACM Trans. Comput. Theory, 2013）此前建立的$\textsf{TC}^{0}(\textsf{FOLL})$上界。