In this paper, we investigate the computational complexity of isomorphism testing for finite groups and quasigroups, given by their multiplication tables. We crucially take advantage of their various decompositions to show the following: - We first consider the class of groups that admit direct product decompositions, where each indecompsable factor is $O(1)$-generated, and either perfect or centerless. We show any group in this class is identified by the $O(1)$-dimensional count-free Weisfeiler--Leman (WL) algorithm with $O(\log \log n)$ rounds, and the $O(1)$-dimensional counting WL algorithm with $O(1)$ rounds. Consequently, the isomorphism problem for this class is in $\textsf{L}$. This improves upon the previous upper bound of $\textsf{TC}^{1}$, which was obtained using $O(\log n)$ rounds of the $O(1)$-dimensional counting WL (Grochow and Levet; FCT 2023, \textit{J. Comput. Syst. Sci.} 2026). - We next consider more generally, the class of groups where each indecomposable factor is $O(1)$-generated. We exhibit an $\textsf{AC}^{3}$ canonical labeling procedure for this class. Here, we accomplish this by showing that in the multiplication table model, the direct product decomposition can be computed in $\textsf{AC}^{3}$, parallelizing the work of Kayal and Nezhmetdinov (ICALP 2009). - Isomorphism testing between a central quasigroup $G$ and an arbitrary quasigroup $H$ is in $\textsf{NC}$. Here, we take advantage of the fact that central quasigroups admit an affine decomposition in terms of an underlying Abelian group. Only the trivial bound of $n^{\log(n)+O(1)}$-time was previously known for isomorphism testing of central quasigroups.

翻译：本文研究了由乘法表给出的有限群与拟群的同构判定计算复杂性。我们关键性地利用了它们的各种分解来证明以下结果：- 我们首先考虑允许直积分解的群类，其中每个不可分解因子是$O(1)$生成的，且要么是完美群要么是无中心群。我们证明该类中任意群可由$O(\log \log n)$轮$O(1)$维无计数Weisfeiler--Leman（WL）算法以及$O(1)$轮$O(1)$维计数WL算法识别。因此，该群类的同构问题属于$\textsf{L}$。这改进了先前使用$O(1)$维计数WL算法$O(\log n)$轮得到的$\textsf{TC}^{1}$上界（Grochow与Levet；FCT 2023, \textit{J. Comput. Syst. Sci.} 2026）。- 我们进一步考虑更一般的群类，其中每个不可分解因子是$O(1)$生成的。我们为该类构造了一个$\textsf{AC}^{3}$规范标定过程。此处我们通过证明在乘法表模型中，直积分解可在$\textsf{AC}^{3}$中计算来实现，这并行化了Kayal与Nezhmetdinov（ICALP 2009）的工作。- 中心拟群$G$与任意拟群$H$间的同构判定属于$\textsf{NC}$。这里我们利用了中心拟群允许基于底层阿贝尔群的仿射分解这一性质。此前对于中心拟群同构判定仅知平凡上界$n^{\log(n)+O(1)}$时间。