Learning to be Simple

TL;DR

The paper addresses the problem of determining when a 2-generated subgroup of the symmetric group $S_n$ is simple, using machine learning to uncover informative generator features. It compares representations ranging from full permutation generators to traces, determinants, and group orders, demonstrating high predictive accuracy and uncovering structure in the data. A machine-guided conjecture emerges from these results and is then proven, yielding a theorem that constrains generator properties (determinants and traces) for finite simple groups and a corollary on fixed-point ratios that holds across sporadic groups. This work showcases the potential of machine-assisted reasoning to inspire rigorous mathematical theorems and offers a framework for exploring vast algebraic datasets with practical implications for pure mathematics.

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.

Figures (6)

• Figure 1: Validation accuracy for varied amount of datasets for each $n$. Percentages are of the total dataset, after balancing but before splitting to allow $k$-fold cross validation.
• Figure 2: Loss and accuracy on training data while training on full datasets, for $n \in \{ 5, 6, 7 \}$. Bold curves give the average value at that epoch across all cross-validation runs.
• Figure 3: Error in predictions of accuracy from simplicity (dashed) versus parity (solid) during training for $n = 5$ (left) and $n = 6$ (right).
• Figure 4: A classifier predicts simplicity of a group based on orders and traces of group generators and the property of being Abelian. The plot shows a confusion plot where the class $0$ stands for simple groups and $1$ for nonsimple groups. Using a neural network model and $1000$ training points, class accuracies were found to be $\approx 97\%$ and $85\%$, respectively, on the training set (left) and $\approx 97\%$ and $86\%$ on the test set (right). We use $1000$ (top row) and $4000$ (bottom row) points for training.
• Figure 5: A classifier predicts simplicity of a group based on orders of group elements and order of the group. The plot shows a confusion plot where the class $0$ stands for simple groups and $1$ for nonsimple groups. Using a neural network model when using only $100$ training points (top) and $1000$ training points (bottom), class accuracies on the test set were found to be $\approx 99\%$.

Theorems & Definitions (5)

• Theorem 1
• Definition 1
• Proposition 1
• proof
• Corollary 1