Date of Award

3-2018

Document Type

Thesis

Degree Name

Master of Arts in Mathematics

Department

Mathematics

First Reader/Committee Chair

Zahid Hasan

Abstract

The main objective of this project is to find the original symmetric presentations of some very important finite groups and to give our constructions of some of these groups. We have found the Mathieu sporadic group M11, HS × D5, where HS is the sporadic group Higman-Sim group, the projective special unitary group U(3; 5) and the projective special linear group L2(149) as homomorphic images of the monomial progenitors 11*4 :m (5 :4), 5*6 :m S5 and 149*2 :m D37. We have also discovered 24 : S3 × C2, 24 : A5, (25 : S4), 25 : S3 × S3, 33 : S4 × C2, S6, 29: PGL(2,7), 22 • (S6 : S6), PGL(2,19), ((A5 : A5 × A5) : D6), 6 • (U4(3): 2), 2 • PGL(2,13), S7, PGL (2,8), PSL(2,19), 2 × PGL(2,81), 25 : (S6 × A5), 26 : S4 × D3, U(4,3), 34 : S4, 32 :D6, 2 • (PGL(2,7) :PSL(2,7), 22 : (S5 : S5) and 23 : (PSL3(4) : 2) as homomorphic images of the permutation progenitors 2*8 : (2 × 4 : 2), 2*16: (2 × 4 :C2 × C2), 2*9: (S3 × S3), 2*9: (S3 × A3), 2*9: (32 × 23) and 2*9: (33 × A3). We have also constructed 24: S3 × C2, 24 : A5, (25: S4), 25 : S3 × S3,: 33: S4 × C2, S6, M11 and U (3,5) by using the technique of double coset enumeration. We have determined the isomorphism types of the most of the images mentioned in this thesis. We demonstrate our work for the following examples: 34 : (32 * 23) × 2, 29 : PGL(2,7), 2•S6, (54 : (D4 × S3)), and 3: •PSL(2,19) ×2.

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