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.
