The purpose of this thesis is to develop original symmetric presentations of finite non-abelian simple groups, particularly the sporadic simple groups. We have found original symmetric presentations for the Janko group J1, the Mathieu group M12, the Symplectic groups S(3,4) and S(4,5), a Lie type group Suz(8), and the automorphism group of the Unitary group U(3,5) as homomorphic images of the progenitors 2*60 : (2 x A5), 2*60 : A5, 2*56 : (23 : 7), and 2*28 : (PGL(2,7):2), respectively. We have also discovered the groups 24 : A5, 34 : S5, PSL(2,31), PSL(2,11), PSL(2,19), PSL(2,41), A8, 34 : S5, A52, 2• A52, 2 : A62, PSL(2,49), 28 : A5, PGL(2,19), PSL(2,71), 24 : A5, 24 : A6, PSL(2,7), 3 x PSL(3,4), 2• PSL(3,4), PSL(3,4), 2• (M12 : 2), 37:S7, 35 : S5, S6, 25 : S6, 35 : S6, 25 : S5, 24 : S6, and M12 as homomorphic images of the permutation progenitors 2*60 : (2 x A5), 2*60 : A5, 2*21 : (7: 3), 2*60 : (2 x A5), 2*120 : S5, and 2*144 : (32 : 24). We have given original proof of the 2*n Symmetric Presentation Theorem. In addition, we have also provided original proof for the Extension of the Factoring Lemma (involutory and non-involutory progenitors). We have constructed S5, PSL(2,7), and U(3,5):2 using the technique of double coset enumeration and by way of linear fractional mappings. Furthermore, we have given proofs of isomorphism types for 7 x 22, U(3,5):2, 2•(M12 : 2), and (4 x 2) :• 22.

The aim of this thesis is to generate original symmetric presentations for finite non-abelian simple groups. We will discuss many permutation progenitors, including but not limited to 2*14 : D28, 2∗9 : 3•(32), 3∗9 : 3•(32), 2∗21 : (7X3) : 2 as well as monomial progenitors, including 7∗5 :m A5, 3∗5 :m S5. We have included their homomorphic images which include the Mathieu group M12, 2•J2, 2XS(4, 5), as well as, many PGL′s, PSL′s and alternating groups. We will give proofs of the isomorphism types of each progenitor, either by hand using double coset enumeration or computer based using MAGMA. We have also constructed Cayley graphs of the following groups, 25 : S5 over 2∗5 : S5, PSL(2, 8) over 2∗7 : D14, M12 over a maximal subgroup, 2XS5. We have developed a lemma using relations to factor permutation progenitors of the form m∗n : N to give an isomorphism of mn : N . Motivated by Robert T. Curtis’ research, we will present a program using MAGMA that, when given a target finite non-abelian simple group, the program will generate possible control groups to write progenitors that will give the given finite non-abelian simple group. Iwasawa’s lemma is also discussed and used to prove PSL(2, 8) and M12 to be simple groups.

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.

In this thesis, we will give our discovery of original symmetric presentations of several important groups. We have investigated permutation and monomial progenitors 2*8: (23: 22), 2*9: (32: 24), 2*10: (24: (2 × 5)), 5*4:m (23: 22), 7*8:m (32: 24), and 3*5:m (24: (2 × 5)). The finite images of the above progenitors include the Mathieu sporadic group M12, the linear groups L2(8) and L2(13), and the extensions S6 × 2, 28 : .L2(8) , and 27 : .A5. We will show our construction of the four groups S3 , L2(8), L2(13), and S6 × 2 over S3, 22, S3 : 2, and S5, by using the technique of double coset enumeration. We will also provide isomorphism types all of the groups that have appeared as finite homomorphic images. We will show that the group L2(8) does not satisfy the conditions of Iwasawas Lemma and that the group L2(13) is simple by Iwasawas Lemma. We give constructions of M22 × 2 and M22 as homomorphic images of the progenitor S6.