Abstract
<title>Abstract</title> <p>Graph Neural Networks (GNNs) have recently shown promise for predicting algebraic properties of finite groups from their Cayley graph representations. However, it remains unclear whether a common graph-learning approach can successfully learn multiple fundamental algebraic properties. To investigate this question, we employ a unified graph-learning pipeline consisting of a common graph construction procedure, feature representation, training methodology, and GNN architecture, with only the target labeling function varying across classification tasks. We evaluate the proposed approach on three representative algebraic properties: abelianity, nilpotency, and solvability. Experiments were conducted on a benchmark of 176 finite groups drawn from several classical families, with all groups included in each classification task. To address class imbalance, class-weighted cross-entropy loss was employed where appropriate during training. Furthermore, the family PSL(2,q) was reserved exclusively for testing, enabling evaluation of the proposed approach's ability to generalize to previously unseen group families. The best-performing models achieved test balanced accuracies of 1.000, 0.856, and 0.875 for abelianity, nilpotency, and solvability, respectively. Although the same graph-learning pipeline was employed across all tasks, different GNN architectures proved optimal for different algebraic properties, suggesting that the representational complexity required to learn a property depends on its underlying algebraic structure. These findings demonstrate that Cayley graph representations contain sufficient structural information to support learning multiple algebraic properties and highlight the potential of graph representation learning for computational group theory.</p>