Learning to be Simple
This work demonstrates a machine learning approach to generate new conjectures and theorems in pure mathematics, specifically for finite group theory, though it is incremental as it applies existing methods to a new domain.
The authors used shallow neural networks to classify finite simple groups among 2-generated subgroups of symmetric groups, achieving varying accuracies, and derived a new theorem about necessary properties of generators for finite simple groups, proven for a class of sporadic groups.
In this work we employ machine learning to understand structured mathematical data involving finite groups and derive a theorem about necessary properties of generators of finite simple groups. We create a database of all 2-generated subgroups of the symmetric group on n-objects and conduct a classification of finite simple groups among them using shallow feed-forward neural networks. We show that this neural network classifier can decipher the property of simplicity with varying accuracies depending on the features. Our neural network model leads to a natural conjecture concerning the generators of a finite simple group. We subsequently prove this conjecture. This new toy theorem comments on the necessary properties of generators of finite simple groups. We show this explicitly for a class of sporadic groups for which the result holds. Our work further makes the case for a machine motivated study of algebraic structures in pure mathematics and highlights the possibility of generating new conjectures and theorems in mathematics with the aid of machine learning.