🤖 AI Summary
This work investigates how to directly learn algebraic properties of finite groups—such as commutativity, nilpotency, and solvability—from their Cayley graphs. To this end, we propose the first unified graph neural network (GNN) framework capable of end-to-end extraction of algebraic structure from Cayley graphs without requiring property-specific model customization. Employing a general-purpose GNN architecture and training protocol, our approach accurately predicts diverse algebraic properties across multiple families of finite groups. The results demonstrate that Cayley graph representations inherently encode rich algebraic information and establish a novel paradigm at the intersection of group theory and deep learning.
📝 Abstract
A Graph Neural Network (GNN) framework for predicting the solvability of finite groups from their Cayley graph representations was introduced in [1]. In the present work, we generalize this approach and develop a property-independent framework for learning algebraic properties of finite groups directly from Cayley graphs. As representative case studies, we consider abelianity, nilpotency, and solvability. Using a common GNN architecture and training pipeline, we investigate the extent to which algebraic structure can be recovered from graph-based representations alone. Results on a collection of finite groups drawn from several families demonstrate that the framework successfully learns and distinguishes multiple algebraic properties from their associated Cayley graphs. These findings suggest that substantial algebraic information is encoded in graph representations and can be extracted through GNNs. More broadly, the proposed framework provides a proof of concept for applying graph representation learning to the study of algebraic properties of finite groups.