A General Framework for Learning Algebraic Properties from Cayley Graphs using Graph Neural Networks

Research Approach

This paper introduces a property-independent framework that uses Graph Neural Networks (GNNs) to classify finite groups based on their algebraic properties. By representing groups as Cayley graphs—geometric structures that encode group elements and generators—the author investigates whether GNNs can extract sufficient structural information to identify whether a group is abelian, nilpotent, or solvable. The methodology employs a consistent training pipeline and GNN architecture across all tasks, using balanced accuracy to account for potential class imbalances in the datasets.

Key Findings

The framework successfully demonstrates that algebraic properties leave detectable structural signatures in Cayley graphs. For abelianity, the model achieved a test balanced accuracy of 0.929, while solvability and nilpotency reached 0.857 and 0.833, respectively. A critical aspect of the evaluation was the inclusion of the Projective Special Linear (PSL) group family in the test set, which was entirely excluded from training. The model's ability to correctly classify these unseen groups suggests that the GNNs are learning generalizable structural features rather than simply memorizing family-specific patterns.

This work provides a proof of concept for using graph representation learning to automate the study of algebraic structures. By showing that GNNs can recover abstract group-theoretic properties from graph representations, the paper bridges the gap between machine learning and pure mathematics. This approach offers a potential pathway for exploring larger or more complex algebraic structures where manual classification or traditional computational methods might be computationally prohibitive.