Graph Neural Networks for Learning Algebraic Properties of Finite Groups from Cayley Graphs

Technical Summary: Graph Neural Networks for Learning Algebraic Properties of Finite Groups from Cayley Graphs

Problem Statement
While Graph Neural Networks (GNNs) have demonstrated success in extracting structural information from graph-structured data, it remains unclear whether a unified graph-learning approach can successfully learn multiple fundamental algebraic properties of finite groups simultaneously. Previous work established that GNNs could predict solvability from Cayley graphs, but it is unknown if Cayley graph representations contain sufficient structural information to support the learning of other core properties, such as abelianity and nilpotency, under a common framework. This paper investigates whether a single graph-learning pipeline, varying only in the target labeling function, can generalize across these distinct algebraic hierarchies.

Methodology
The author proposes a property-independent framework that treats finite groups as independent data instances represented by their Cayley graphs. The methodology consists of four unified stages:

1. Group Generation: A benchmark of 176 finite groups was constructed, comprising representatives from classical families (Cyclic, Dihedral, Quaternion, Symmetric, Alternating, Projective Special Linear) and numerous direct-product constructions. This diversity ensures the model learns structural features rather than family-specific heuristics.
2. Cayley Graph Construction: For each group, an undirected Cayley graph was generated using fixed, symmetric generating sets appropriate to the group family (e.g., rotation/reflection for Dihedral groups, transpositions/cycles for Symmetric groups).
3. Target Labeling: Each graph was assigned a binary label corresponding to one of three classification tasks: Abelian vs. non-Abelian, Nilpotent vs. non-Nilpotent, and Solvable vs. non-Solvable.
4. GNN Modeling: The framework employs a family of GNN architectures based on Graph Convolutional Network (GCN) layers. Models consist of two GCNConv layers with GELU activation, followed by global mean pooling to generate graph-level embeddings. These are passed through fully connected layers for binary classification.
   • Training Strategy: To address class imbalance, class-weighted cross-entropy loss was used. The Adam optimizer (learning rate 0.002) trained models for 100 epochs.
   • Generalization Protocol: A rigorous hold-out strategy was employed where the entire family of Projective Special Linear groups, $PSL(2, q)$, was excluded from training and validation, reserved exclusively for testing to evaluate generalization to unseen algebraic structures.
   • Evaluation: Performance was measured using Balanced Accuracy (BA) to account for uneven class distributions.

Key Contributions
The paper makes four primary contributions:

1. Unified Learning: It demonstrates that multiple fundamental algebraic properties (abelianity, nilpotency, solvability) can be learned directly from Cayley graph representations using a common graph-learning pipeline.
2. Comparative Evaluation: It evaluates these three properties using identical graph construction, feature representation, and training methodologies, isolating the target property as the sole variable.
3. Expanded Benchmark: It introduces an expanded dataset of 176 finite groups, significantly larger and more diverse than previous benchmarks, including direct-product constructions.
4. Generalization Evidence: It provides empirical evidence of generalization to the previously unseen $PSL(2, q)$ family, which was entirely excluded from the training phase.

Results
The experiments yielded the following test Balanced Accuracy (BA) scores for the best-performing architectures:

• Abelianity: 1.000 (Perfect classification). The optimal architecture was a simple (2-16-16) hidden dimension network.
• Nilpotency: 0.856. The optimal architecture required higher capacity (2-64-64).
• Solvability: 0.875. The optimal architecture was the simplest (2-8-8).

Notably, different GNN architectures proved optimal for different properties, suggesting that the representational complexity required to learn a property depends on its underlying algebraic structure. Crucially, the models correctly classified all or most of the held-out $PSL(2, q)$ groups, indicating that the learned representations capture general structural signatures rather than memorizing specific group families.

Significance and Claims
The paper positions this work as a "proof of concept" demonstrating that Cayley graph representations encode sufficient structural information to support the learning of multiple algebraic properties without property-specific feature engineering. The author claims that these findings highlight the potential of graph representation learning for computational group theory.

The study concludes that a unified graph-learning framework can successfully predict abelianity, nilpotency, and solvability, contributing to the intersection of group theory, graph theory, and geometric deep learning. The author remains modest regarding the scope, acknowledging limitations such as the small subset of finite groups tested, the reliance on specific generating sets, and the need for further investigation into additional algebraic properties and larger benchmarks. They do not claim to have solved the general isomorphism problem or to have replaced traditional algebraic algorithms, but rather to have established a viable ML-based approach for property prediction from structural data.