ABSTRACT OF THE THESIS

Computational Tools for Group Theory

by

Meenal Garg

Master of Science in Computer Science

This thesis describes the refinement and extension of code that was originally developed as part of a 1987 Math 797 project by David Gibbs, “Computer Generation and Identification of Groups of Order 2 to 31” by Jeff Barr. His thesis was able to generate all groups of order 2-31 as well as few groups of order 32. He was able to calculate inner automorphism of the groups. He also allowed a user to manually enter a Cayley Table. The purpose of this code was to enhance the interface which Jeff Barr provided along with extending the identification of the groups of the order from 32-63. This interface was also able to generate, identify, and analyze groups presented in the form of a Cayley Table. Objects were created to improve the code design and allow for better interaction between the generation, identification, analysis, and visualization sections of code. The code for this thesis allows cyclic groups to easily be generated, along with groups created via defined relationships, semi direct products and the cross product of multiple groups. A user interface was improved upon to the system to assist the user when utilizing the code as well as visualizing the groups that are generated. It can now allow user to change the values which they added before. The interface is more helpful for the users who are new to group theory as well as computer.

The generation code is no longer limited to groups of order less than 31. It can identify all the groups of order 32-63. It can identify few groups of order 64 as well. Improvements were made to the identification code so that the system can identify all Abelian groups including those created via the cross product of group and semi direct product groups. Additionally, many non-Abelian groups of order 32 were added to the list of groups which Jeff Barr’s thesis could not identify. Group Analysis functionality was added including the identification of whether a table actually represents a group as well as if the group is commutative, has identity or has inverse. It also lists the centers of the groups on the interface.

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