Non-Abelian Finite Simple Groups as Homomorphic Images

Author

Sandra BahenaFollow

Date of Award

8-2021

Document Type

Thesis

Degree Name

Master of Arts in Mathematics

Department

Mathematics

First Reader/Committee Chair

Hasan, Zahid

Abstract

The purpose of exploring infinite groups in this thesis was to discover homomorphic images of non-abelian finite simple groups. These infinite groups are semi-direct products known as progenitors. The permutation progenitors studied were: 2^*8 ∶ 2^{2 ∙} A₄, 2^*10 ∶ D₂₀, 2^*4 ∶ C₄, 2^*7 ∶ (7 ∶ 6), 3^*3 ∶ S₃, 2^*15 ∶ ((5 × 3) ∶ 2), and 2^*20 ∶ A₅. When we factored said progenitors by an appropriate number of relations, we produced several original symmetric presentations and constructions of linear groups, other classical groups, and sporadic groups. We have found original symmetric presentations of several important groups, including: PGL₂(7), PSL₂(8), PSL₂(11), PGL₂(11), PGL₂(13), PSL₂(19), PGL₂(29), PSL₂(41), PSL₂(71), J₂, U₃(4), U₃(5), M₁₁, and M₂₂. When solving various extension problems, we are able to identify the isomorphism types of the finite images we discovered. We present proofs of the isomorphism types of the finite images that were found by solving extension problems. The four types of extension problems are Direct Product, Semi-Direct Product, Central Extensions, and Mixed Extensions. We perform double coset enumeration manually and with the support of a computer-based program, Magma, to construct Cayley diagrams of 3² ∶ S₃, M₁₁, PSL₂(19), PSL₂(7), S₄, and U₃(5) ∶ 2.

Recommended Citation

Bahena, Sandra, "Non-Abelian Finite Simple Groups as Homomorphic Images" (2021). Electronic Theses, Projects, and Dissertations. 1308.
https://scholarworks.lib.csusb.edu/etd/1308