On the parallel complexity of group isomorphism via Weisfeiler–Leman

We leverage the Weisfeiler–Leman algorithm for groups (Brachter & Schweitzer, LICS 2020) to improve parallel complexity upper bounds on isomorphism testing for several families of groups. We first show that groups with an Abelian normal Hall subgroup whose complement is $O\left(1\right)$-generated are identified by constant-dimensional Weisfeiler–Leman using $O\left(1\right)$-rounds. This places isomorphism testing for this family of groups into $\text{L}$; the previous upper bound for isomorphism testing was $P$ (Qiao, Sarma, & Tang, STACS 2011). We next use the individualize-and-refine paradigm to obtain an isomorphism test for groups without Abelian normal subgroups by $\mathrm{SAC}$ circuits of depth $O(\log n)$ and size $n^{O(\log \log n)}$, previously only known to be in $P$ (Babai, Codenotti, & Qiao, ICALP 2012) and ${\mathrm{quasiSAC}}^1$ (Chattopadhyay, Torán, & Wagner, ACM Trans. Comput. Theory, 2013). We next extend a result of Brachter & Schweitzer (ESA, 2022) on direct products of groups to the parallel setting. Namely, we how 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 $P$. 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 ${\beta }_1{\text{MAC}}^0\left(\text{FOLL}\right)$ for isomorphism testing of Abelian groups. This improves upon the previous ${\text{TC}}^0\left(\text{FOLL}\right)$ upper bound due to Chattopadhyay, Torán, & Wagner (ACM Trans. Comput. Theory, 2013).