Homomorphic Images And Related Topics

iii We will explore progenitors extensively throughout this project. The progen-itor, developed by Robert T Curtis, is a special type of infinite group formed by a semidirect product of a free group m∗n and a transitive permutation group of degree n. Since progenitors are infinite, we add necessary relations to produce finite homomorphic images. Curtis proved that any non-abelian simple group is a homomorphic image of a progenitor of the form 2∗n: N. In particular, we will investigate progenitors that generate two of the Mathieu sporadic groups, M11 and M22, as well as some classical groups. We will prove their existences a variety of different ways, including the process of double coset enumeration, Iwasawa’s Lemma, and linear fractional mappings. We will also investigate the various techniques of finding finite images and their corresponding isomorphism types. iv Acknowledgement