Neural Network Based Classifier for Some Permutation Groups

Abstract

This thesis explores the membership problem within the Symmetric group, specifi cally focusing on determining whether a randomly generated permutation in Symmetric group Sn is a member of Sn . For computational convenience, the study centers on the case where n=5, employing a Neural Network implemented in the Python programming language. Initially, a single perceptron with one neuron was developed. After multiple itera tions of the neural network, accuracy levels ranging from 60% to 85% were achieved. Subsequently, the investigation advanced to a multi-layered neural network. Utilizing this sophisticated architecture, the model was trained to identify the nature of specific permutations, distinguishing between even or odd permutations and determining their membership in S5 . The desired accuracy levels were successfully reached by varying the amounts of data used in the training process.