#### Date of Award

6-2015

#### Document Type

Thesis

#### Degree Name

Master of Arts in Mathematics

#### Department

Mathematics

#### First Reader/Committee Chair

Hasan, Zahid

#### Abstract

The goal of this thesis is to show constructions of some of the sporadic groups such as the Mathieu group, M_{12}, J_{1}, Projective Special Linear groups, PSL(2,8), and PSL(2,11), Unitary group U(3,3) and many other non-abelian simple groups. Our purpose is to find all simple non-abelian groups as homomorphic images of permutation or monomial progenitors, as well grasping a deep understanding of group theory and extension theory to determine groups up to isomorphisms. The progenitor, developed by Robert T. Curtis, is a semi-direct product of the following form: P≅2*^{n}: N = {πw | π ∈ N, w a reduced word in the t_{i}} where 2*^{n} denotes a free product of n copies of the cyclic group of order 2 generated by involutions t_{i }for 1 ≤ i≤ n; and N is a transitive permutation group of degree n which acts on the free product by permuting the involuntary generators by conjugation. Thus we develop methods for factoring by a suitable any number of relations in the hope of finding all non-abelian simple groups, and in particular one of the 26 Sporadic simple groups. Then the algorithm for double coset enumeration together with the first isomorphic theorem aids us in proving the homomorphic image of the group we have constructed. After being presented with a group G, we then compute the composition series to solve extension problems. Given a composition such as G = G0 ≥ G_{1} ≥ ….. ≥ G_{n-1} ≥ G_{n} = 1 and the corresponding factor groups G_{0}/G_{1} = Q_{1},…,G_{n-2}/G_{n-1 }= Q_{n-1},G_{n-1}/G_{n} = Q_{n}. We note that G_{1} = 1, implying G_{n-1} = Q_{n}. As we move through the next composition factor we see that G_{n-2}/Q_{n} = Q_{n-1}, so that G_{n-2 }is an extension of Q_{n-1} by Q_{n}. Following this procedure we can recapture G from the products of Q_{i} and thus solve the extension problem. The Jordan-Holder theorem then allows us to develop a process to analyze all finite groups if we knew all finite simple groups and could solve their extension problem, hence arriving at the isomorphism type of the group. We will present how we solve extensions problems while our main focus will lie on extensions that will include the following: semi-direct products, direct products, central extensions and mixed extensions.Lastly, we will discuss Iwasawa's Lemma and how double coset enumeration aids us in showing the simplicity of some of our groups.

#### Recommended Citation

Lamp, Leonard B., "SYMMETRIC PRESENTATIONS OF NON-ABELIAN SIMPLE GROUPS" (2015). *Electronic Theses, Projects, and Dissertations*. 222.

https://scholarworks.lib.csusb.edu/etd/222