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In this thesis, we have presented our discovery of true finite homomorphic images of various permutation and monomial progenitors, such as 2*7: D14, 2*7 : (7 : 2), 2*6 : S3 x 2, 2*8: S4, 2*72: (32:(2S4)), and 11*2 :m D10. We have given delightful symmetric presentations and very nice permutation representations of these images which include, the Mathieu groups M11, M12, the 4-fold cover of the Mathieu group M22, 2 x L2(8), and L2(13). Moreover, we have given constructions, by using the technique of double coset enumeration, for some of the images, including M11 and M12. We have given proofs, either by hand or computer-based, of the isomorphism type of each image. In addition, we use Iwasawa's Lemma to prove that L2(13) over A5, L2(8) over D14, L2(13) over D14, L2(27) over 2D14, and M11 over 2S4 are simple groups. All of the work presented in this thesis is original to the best of our knowledge.

S(5,8,24) is one of the largest known Steiner systems and connects combinatorial designs, error-correcting codes, finite simple groups, and sphere packings in a truly remarkable way. This thesis discusses the underlying structure of S(5,8,24), its construction via the (24,12) Golay code, as well its automorphism group, which is the Mathieu group M24, a member of the sporadic simple groups. Particular attention is paid to the calculation of the size of automorphism groups of Steiner systems using the Orbit-Stabilizer Theorem. We conclude with a section on the sphere packing problem and elaborate on how the 8-sets of S(5,8,24) can be used to form Leech's Lattice, which Leech used to create the densest known sphere packing in 24-dimensions. The appendix contains code written for Matlab which has the ability to construct the octads of S(5,8,24), permute the elements to obtain isomorphic S(5,8,24) systems, and search for certain subsets of elements within the octads. / Master of Science

[pt] Os cinco grupos de Mathieu, M24;M23;M22;M12 e M11, compõem a primeira família de grupos esporádicos do Teorema de classificacão dos grupos simples finitos. Neste trabalho apresentaremos os grupos de Mathieu e alguns objetos relacionados a construcão deles como o Código de Golay e o Sistema de Steiner. Também, no decorrer do texto, surgiram espontaneamente alguns subgrupos dos grupos de Mathieu. / [en] The five Mathieu groups, M24;M23;M22;M12 and M11, form the first family of sporadic groups of Theorem classification of finite simple groups. In this paper we present the Mathieu groups and some objects related to building them as the Golay code and the Steiner system. Also, throughout the text, arose spontaneously some subgroups of groups of Mathieu.