Learning to be simple: machine learning uncovers structures in finite simple groups

A research team of mathematicians and computer scientists has used machine learning to reveal new mathematical structure within the theory of finite groups. By training neural networks to recognise simplicity in algebraic data, the team discovered and proved a new theorem on the necessary properties of generators of finite simple groups. This work demonstrates how artificial intelligence can assist in formulating and even proving conjectures in pure mathematics. The 2-generator representation furthers earlier work of one of the authors with M. Kim using Cayley Tables, showing that simplicity has interesting data structure.

Finite simple groups are the “atoms” of group theory—building blocks of all finite groups. Despite their central role in algebra, identifying simplicity from a presentation is difficult. In this study, the researchers constructed a comprehensive database of all 2-generated subgroups of the symmetric group on n objects and trained shallow feed-forward neural networks to classify which are simple. The model successfully learned to distinguish simple groups with high accuracy, depending on the choice of input features.

The Solution: Analysis of the trained networks revealed patterns that pointed to a specific property of group generators. From these observations, a conjecture emerged on the necessary conditions for the generators of any finite simple group. The conjecture was subsequently proven, forming a new “toy theorem” that holds for several families, including certain sporadic groups.

The Future: Future research will expand this data-driven framework to explore further conjectures relating to the representation of simple groups and finite groups in general.

The Impact: This study marks a step toward a paradigm where artificial intelligence contributes directly to mathematical discovery. By bridging machine learning and group theory, it showcases how computation can illuminate abstract structure and inspire rigorous new theorems in representation theory.

Reference: Yang-Hui He, Vishnu Jejjala, Challenger Mishra, and Em Sharnoff. Learning to be Simple[J]. AI For Science . DOI: 10.1088/3050-287X/ae1d98

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