Investigation of Finite Groups Through Progenitors

Author

Charles BaccariFollow

Date of Award

12-2017

Document Type

Thesis

Degree Name

Master of Arts in Mathematics

Department

Mathematics

First Reader/Committee Chair

Zahid Hasan

Abstract

The goal of this presentation is to find original symmetric presentations of finite groups. It is frequently the case, that progenitors factored by appropriate relations produce simple and even sporadic groups as homomorphic images. We have discovered two of the twenty-six sporadic simple groups namely, M₁₂, J₁ and the Lie type group Suz(8). In addition several linear and classical groups will also be presented. We will present several progenitors including: 2^*12: 2² x (3 : 2), 2^*11: PSL₂(11), 2^*5: (5 : 4) which have produced the homomorphic images: M₁₂ : 2, Suz(8) x 2, and J₁ x 2. We will give monomial progenitors whose homomorphic images are: 17^*10 :_m PGL₂(9), 3^*4:_m Z₂ ≀D₄ , and 13^*2:_m (2² x 3) : 2 which produce the homomorphic images:13² : ((2 x 13) : (2 x 3)), 2 x S_9, and (2²)^•PGL₄(3). Once we have a presentation of a group we can verify the group's existence through double coset enumeration. We will give proofs of isomorphism types of the presented images: S₃ x PGL₂(7) x S₅, 2⁸:A₅, and 2^•U₄(2):2.

Recommended Citation

Baccari, Charles, "Investigation of Finite Groups Through Progenitors" (2017). Electronic Theses, Projects, and Dissertations. 600.
https://scholarworks.lib.csusb.edu/etd/600