Publication Date

Spring 5-12-2025

Document Type

Thesis

Degree Name

Master of Science in Mathematics (MS)

Department

Mathematics

Committee Chair

Rohitha Goonatilake

Committee Member

Cesar Contreras

Committee Member

Runchang Lin

Committee Member

Norma Saikali

Abstract

A foundational idea in mathematics lies in breaking down existing components into their bare fundamentals. As evidenced by prime numbers and composites, we learn this idea at an early age. Categorizing these broken-down components into their simplest form allows mathematicians to construct proofs from emergent patterns. John Conway’s Atlas of Finite Groups in the 1990s was particularly concerned with the categorization of structures known as groups. There are certain axioms a group must adhere to, which amount to the retention of symmetry; ultimately a group helps us to better understand symmetric actions performed on a set with a binary operation.

The Jordan-H¨older Theorem states that a finite group, that is, a binary operation coupled with a set, which contains a finite amount of symmetries, will break down into a simpler group; these are known as follows: cyclic groups, alternating groups, groups of Lie type, or a sporadic group (of which there are 26 varieties). This classification of finite groups breaking down into simple finite groups is considered a milestone in modern mathematics. The goal of this research is to search for applications of sporadic groups. It is our hope that we will yield applications for sporadic groups, in particular Mathieu groups in other fields of mathematics and sciences.

Deoxyribonucleic acid (DNA) sequencing is one potential application for the use of these sporadic groups. Mathieu groups have existing applications in error correction because they are automorphisms of a linear code. Error correction ensures that the transmission of data is consistently possible and that noise in a system cannot corrupt the information in it; the goal is to discover if a similar approach might allow for error correction in mutations.

It is assumed that by treating the set of DNA sequences as a linear code, we may iv be able to apply error correcting algorithms to preserve intended sequences of DNA and prevent harmful mutations. A set of codewords that form a vector subspace of a finite vector space over a finite field is called a linear code; treating DNA as such is one assumption made in the research. We will also discover if perhaps DNA may already come equipped with such properties.

The study of certain open problems in mathematics allows us to sometimes put to the test existing theorems and discover new applications. It is our aim that through the work we may discover a sequence, encoding function or group isomorphism that can emulate DNA sequencing, for the medical technology to flourish.

Throughout the course of the research, we have shown encodings and decodings of both DNA sequences applied through Hamming and Golay codes, but discovered that a group acting as an automorphism to these codes does not necessarily act as an automorphism on the DNA code. We consider the need for other functions to accommodate for the intricacies imposed by the geometry of nucleotides and enzymes.

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