We will explore progenitors extensively throughout this project. The progenitor, developed by Robert T Curtis, is a special type of infinite group formed by a semi-direct 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 found 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 M11, 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.

The goal of this thesis is to show constructions of some of the sporadic groups such as the Mathieu group, M12, J1, Projective Special Linear groups, PSL(2,8), and PSL(2,11), Unitary group U(3,3) and many other non-abelian simple groups. Our purpose is to find all simple non-abelian groups as homomorphic images of permutation or monomial progenitors, as well grasping a deep understanding of group theory and extension theory to determine groups up to isomorphisms. The progenitor, developed by Robert T. Curtis, is a semi-direct product of the following form: P≅2*n: N = {πw | π ∈ N, w a reduced word in the ti} where 2*n denotes a free product of n copies of the cyclic group of order 2 generated by involutions ti for 1 ≤ i≤ n; and N is a transitive permutation group of degree n which acts on the free product by permuting the involuntary generators by conjugation. Thus we develop methods for factoring by a suitable any number of relations in the hope of finding all non-abelian simple groups, and in particular one of the 26 Sporadic simple groups. Then the algorithm for double coset enumeration together with the first isomorphic theorem aids us in proving the homomorphic image of the group we have constructed. After being presented with a group G, we then compute the composition series to solve extension problems. Given a composition such as G = G0 ≥ G1 ≥ ….. ≥ Gn-1 ≥ Gn = 1 and the corresponding factor groups G0/G1 = Q1,…,Gn-2/Gn-1 = Qn-1,Gn-1/Gn = Qn. We note that G1 = 1, implying Gn-1 = Qn. As we move through the next composition factor we see that Gn-2/Qn = Qn-1, so that Gn-2 is an extension of Qn-1 by Qn. Following this procedure we can recapture G from the products of Qi and thus solve the extension problem. The Jordan-Holder theorem then allows us to develop a process to analyze all finite groups if we knew all finite simple groups and could solve their extension problem, hence arriving at the isomorphism type of the group. We will present how we solve extensions problems while our main focus will lie on extensions that will include the following: semi-direct products, direct products, central extensions and mixed extensions.Lastly, we will discuss Iwasawa's Lemma and how double coset enumeration aids us in showing the simplicity of some of our groups.