#### Date of Award

12-2018

#### Document Type

Thesis

#### Degree Name

Master of Arts in Mathematics

#### Department

Mathematics

#### First Reader/Committee Chair

Hasan, Zahid

#### Abstract

In this project, we demonstrate our discovery of original symmetric presentations and constructions of important groups, including nonabelian simple groups, and groups that have these as factor groups. The target nonabelian simple groups include alternating, linear, and sporadic groups. We give isomorphism types for each finite homomorphic image that has been found. We present original symmetric presentations of $M_{12}$, $M_{21}:(2 \times 2)$, $L_{3}(4):2^2$, $2:^{\cdot}L_{3}(4):2$, $S(4,3)$, and $S_{7}$ as homomorphism images of the progenitors $2^{*20}$ $:$ $A_{5}$, $2^{*10}$ $:$ $PGL(2,9)$, $2^{*10}$ $:$ $Aut(A_{6})$, $2^{*10}$ $:$ $A_{6}$, $2^{*10}$ $:$ $A_{5}$, and $2^{*24}$ $:$ $S_{5}$, respectively. We also construct $M_{12}$, $M_{21}:(2 \times 2)$, $L_{3}(4):2^2$, $L_{3}(4):2^2$, $2:^{\cdot}L_{3}(4):2$, $S(4,3)$, and $S_{7}$ over $A_{5}$, $PGL(2,9)$, $Aut(A_{6})$, $A_{6}$, $A_{5}$, and $S_{5}$, respectively, using our technique of double coset enumeration. All of the symmetric presentations given are original to the best of our knowledge.

#### Recommended Citation

Seager, Charles, "Symmetric Presentations and Double Coset Enumeration" (2018). *Electronic Theses, Projects, and Dissertations*. 783.

https://scholarworks.lib.csusb.edu/etd/783