On the Parallel Complexity of Identifying Groups and Quasigroups via Decompositions

TL;DR

The paper investigates the parallel complexity of isomorphism testing for finite groups and quasigroups given by multiplication tables, leveraging direct product decompositions and Weisfeiler--Leman (WL) techniques. It shows that for the class $\\mathcal{C}$ of groups with fully refined direct product decompositions into indecomposables that are $O(1)$-generated and either perfect or centerless, count-free WL Version II with $O(\log\log n)$ rounds and counting WL Version II with $O(1)$ rounds identify the group, placing isomorphism testing for this class in $\\mathsf{L}$ and improving earlier $\\mathsf{TC}^1$ bounds. It also extends to a broader class $\\mathcal{D}$ with an $\\mathsf{AC}^3$ canonization procedure and proves that isomorphism testing between a central quasigroup and an arbitrary quasigroup lies in $\\mathsf{NC}$, using an affine decomposition over an underlying abelian group and transporter-based reductions. Methodologically, the work combines parallel WL analyses, direct product decomposition via Kayal–Nezhmetdinov style parallelization, and central-quasigroup structural reductions to obtain tighter parallel complexity bounds and constructive canonical forms. The results advance understanding of when group/quasigroup isomorphism can be solved efficiently in parallel, and provide practical routes for canonization and isomorphism testing in structured algebraic families with decomposable architectures.

Abstract

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.

Theorems & Definitions (73)

• Definition 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Corollary 1.5
• Theorem 1.6
• Theorem 1.7: cf. KayalNezhmetdinov
• Remark 1.8
• Definition 1.9
• Theorem 1.10