Date of Award
Master of Arts in Mathematics
First Reader/Committee Chair
A progenitor is an infinite semi-direct product of the form m∗n : N, where N ≤ Sn and m∗n : N is a free product of n copies of a cyclic group of order m. A progenitor of this type, in particular 2∗n : N, gives finite non-abelian simple groups and groups involving these, including alternating groups, classical groups, and the sporadic group. We have conducted a systematic search of finite homomorphic images of numerous progenitors. In this thesis we have presented original symmetric presentations of the sporadic simple groups, M12, J1 as homomorphic images of the progenitor 2∗12 : (2×A5), M22 and M22 : 2 as homomorphic images of 2∗14 : (23 : 7) and J2 as a homomorphic image of 2∗160 : PSL(2, 11). We have also given original symmetric presentations of a number of alternating and classical groups and symmetric groups such as PSL(2, 7), PSL(2, 19), PSL(2, 41), PSL(2, 8), A8, S7, and S8. We have also searched for finite homomorphic images of the monomial progenitor: 23 : 3 :m 23 : A6 and found the original symmetric presentations of the image 26 : Sym(5). We construct the following images by using our technique of double coset enumeration: 34 : (22 : S3) over (33 : (3 : 2)),J1 over (2 × A5),(24) : (S5 : 2) over S5, 73 : S3 : 2 over (21 × (S6),27 : PSL(2, 7) over PSL(2, 7), and 27 : (7 : 3) over (23 : (7 : 3). In addition, we give isomorphism class of each image that we have discovered.
Kasouha, Samar Mikhail, "SYMMETRIC PRESENTATIONS OF FINITE GROUPS AND RELATED TOPICS" (2022). Electronic Theses, Projects, and Dissertations. 1460.