Graph Neural Networks for Predicting Solvability of Finite Groups

This study investigates Graph Neural Networks (GNNs) for predicting the solvability of finite groups represented by Cayley graphs. A two-layer Graph Convolutional Network (GCN) is employed to learn structural information from these graphs and classify groups as solvable or non-solvable. Experiments on 300 groups consider two settings: S-reserved and A-reserved, in which symmetric and alternating group families, respectively, are excluded from both training and validation. Both settings also include groups from non-reserved families in their test sets. Across 40 random seeds, the GNN achieved mean balanced accuracies of 0.9423 and 0.9014, respectively, outperforming Logistic Regression, Decision Tree, and Random Forest classifiers. Paired one-sided t-tests indicated statistically significant differences in performance relative to all three baselines. Five-fold cross-validation was also conducted to evaluate model stability. These findings suggest that GNNs can capture graph-based structural information relevant to finite-group solvability, with promising results on test sets containing previously unseen group families.