Learning to be Simple (2312.05299v1)

Abstract: 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.

• The paper presents a machine learning approach that classifies the simplicity of finite groups using shallow feed-forward neural networks.
• It leverages various representations, including permutation matrices and invariants like traces and determinants, to capture key structural properties.
• The study formulates a new conjecture linking generator properties to group simplicity, ultimately proving a novel theorem in algebra.

Introduction

Machine learning has increasingly become a versatile tool in both applied and theoretical domains, ranging from practical applications such as autonomous vehicles and the design of pharmaceuticals to theoretical explorations in quantum chemistry and biology. Its role in the development of pure mathematics, however, has been relatively limited. Within the sphere of algebraic structures, groups are fundamental, representing the symmetries of a system and playing a critical role in the physical understanding of nature through principles such as Noether's theorem. This paper investigates whether machine learning algorithms can be trained to detect the simplicity of groups – a characteristic indicating the absence of non-trivial normal subgroups within the group's structure.

Machine Learning and Simplicity of Groups

The paper focuses on the characteristic of simplicity within finite groups, which are well-studied in mathematics and can be partially classified into specific categories like cyclic, alternating, or sporadic groups. A database of 2-generated subgroups of symmetric groups was compiled and used for supervised learning with shallow feed-forward neural networks aimed at classifying the simplicity of the groups. This approach leveraged various representations of the groups, including permutation matrices of the generators and specific invariants like traces and determinants. The results varied according to the representation method used, ultimately leading to the formulation of a natural conjecture concerning the generators of finite simple groups.

Learning Outcomes and A Conjecture

The authors report on a series of machine learning experiments conducted with different input features and neural network configurations. Generating a group from its two generators and determining its simplicity traditionally would involve exhaustive and computationally expensive methods. However, the paper demonstrates that a multi-layer perceptron model can predict group simplicity with noteworthy accuracies, especially when using higher-level properties of the generators as features (e.g., orders of group elements). Machine learning outcomes suggest a possible mathematical relationship between generator properties and group simplicity, culminating in the formulation and proof of a new theorem.

Discussion

Undeniably, machine learning can contribute to the recognition of patterns in mathematical datasets undetectable by human analysis. In this paper, machine learning not only unveiled a new theorem regarding the necessary properties of generators for finite simple groups but also showed that a catena of prior computational efforts can be paralleled by the training models, achieving predictive accuracies that confirm the interplay between data representations and underlying mathematics. This work pioneers the engagement of machine learning in pure mathematics, proposing a new path for conjecture generation, and enriches our toolbox for exploring algebraic structures, such as finite simple groups, through artificial intelligence.