🤖 AI Summary
Determining whether a subgroup isomorphism exists between finite groups is a fundamental problem in computational group theory. This work proposes a novel unified learning framework that integrates geometric deep learning with algebraic features: group structures are represented via Cayley graphs, and a siamese graph neural network generates corresponding graph embeddings; these are then combined with handcrafted algebraic invariants to form a joint feature vector, which is fed into a fully connected classifier to predict subgroup relationships. Evaluated on an independent test set, the method achieves an accuracy of 95.9% (47 out of 49 cases), strongly demonstrating the effectiveness and innovation of the proposed framework.
📝 Abstract
Determining whether one finite group is isomorphic to a subgroup of another is a fundamental problem in computational group theory. In this work, we propose a Siamese Graph Neural Network (Siamese GNN) for subgroup prediction using Cayley graph representations of finite groups. Each input group is represented by its undirected Cayley graph and encoded by one branch of a Siamese GNN to produce a graph embedding. The resulting graph embeddings are combined with algebraic features derived directly from the input groups to construct a joint feature vector, which is processed by a fully connected classifier to predict subgroup relations between finite groups. By integrating graph-based structural representations with algebraic features, the proposed framework provides a unified approach for learning subgroup relations from finite groups. Experimental results demonstrate the effectiveness of the proposed architecture, achieving a test accuracy of 95.9% (47/49) on an independent test set and illustrating the potential of geometric deep learning for subgroup prediction.