Symmetric Presentations and Generation

iii The aim of this thesis is to generate original symmetric presentations for nite non-abelian simple groups. We will discuss many permutation progenitors, including 214: D28, 29: (3(32)), 39: (3(32)), 221: ((73) : 2) as well as monomial progenitors, including 75:m A5, 35:m S5. Their homomorphic images include the sporadic Mathieu groupsM11 andM12, the sporadic Janko group 3 J2, the Symplectic group 2S(4; 5), as well as, many linear and alternating groups. We will give proofs of the isomorphism types of each progenitor, either by hand using double coset enumeration or using MAGMA. We have also constructed Cayley diagrams of the following groups, 25: S5 over S5, PGL(3; 4) over M10, PSL(2; 8) over D14 and M12 over the maximal subgroup 2 S5. We have developed a lemma using relations to factor permutation progenitors of the form mn: N to give the group mn: N. We will present a program written in MAGMA that, when given a target nite non-abelian simple group, generates possible progenitors that will giv