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Graph Neural Networks for Predicting Solvability of Finite Groups

We present a Graph Neural Network (GNN) framework for the classification of finite groups according to their solvability. Using undirected Cayley graph representations, the proposed framework learns to distinguish solvable and non-solvable groups directly from structural graph information, without relying on explicit algebraic features. The framework is evaluated on a benchmark dataset of 200 finite groups, comprising 120 solvable and 80 non-solvable groups. The experiments investigate the extent to which GNNs can learn the algebraic property of solvability from Cayley graph representations and generalize to previously unseen finite groups. The selected GNN architecture achieved a balanced accuracy (BA) of 1.000 on the independent test set. Furthermore, repeated experiments using different random seeds and learning rates consistently produced BAs between 0.956 and 1.000, demonstrating the robustness of the proposed framework with respect to the training configuration. To further evaluate generalization, the entire PSL(2,q) family was excluded from the training and validation sets and reserved exclusively for testing. The selected model correctly classified every previously unseen group in this family, demonstrating successful generalization to an entirely unseen family of finite groups.

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