In this thesis, we have presented our discovery of symmetric presentations of a number of non-abelian simple groups, including the Mathieu group M12. We have given several progenitors, permutation and monomial, including 2*4:(22:3), 2*5:D10, 2*8:((4X2).D4), 3*7:m L2(7), 2*6:(Z3 wr Z2), and 2*24: (2. A5) and their homomorphic images which include 4.(M12:2), the group of automorphisms of M12 and several classical groups. We have given the isomorphism type of each of the group mentioned in the thesis. In each case, a proof of the isomorphism type is provided, either computer-based or by hand. In addition, by hand constructions, using the technique of double coset enumeration, are given for the groups L2(11)X3, L2(11), PGL2(11), S5,(A5XA5):4, A7, and 37:L2(7).
","abstract_html":"<p>In this thesis, we have presented our discovery of symmetric presentations of a number of non-abelian simple groups, including the Mathieu group M<sub>12</sub>. We have given several progenitors, permutation and monomial, including 2*<sup>4</sup>:(2<sup>2</sup>:3), 2*<sup>5</sup>:D<sub>10</sub>, 2*<sup>8</sup>:((4X2).D<sub>4</sub>), 3*<sup>7</sup>:<sub>m</sub> L<sub>2</sub>(7), 2*<sup>6</sup>:(Z<sub>3</sub> wr Z<sub>2</sub>), and 2*<sup>24</sup>: (2<sup>.</sup> A<sub>5</sub>) and their homomorphic images which include 4.(M<sub>12</sub>:2), the group of automorphisms of M<sub>12</sub> and several classical groups. We have given the isomorphism type of each of the group mentioned in the thesis. In each case, a proof of the isomorphism type is provided, either computer-based or by hand. In addition, by hand constructions, using the technique of double coset enumeration, are given for the groups L<sub>2</sub>(11)X3, L<sub>2</sub>(11), PGL<sub>2</sub>(11), S<sub>5</sub>,(A<sub>5</sub>XA<sub>5</sub>):4, A<sub>7</sub>, and 3<sup>7</sup>:L<sub>2</sub>(7).</p>","abstract_has_math":false,"creators":["Alharbi, Mashael U"],"institution":null,"degree_name":"Master of Arts in Mathematics","degree_level":"Thesis","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Hasan, Zahid"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2015,"date_issued":"2015-03-01T08:00:00Z","date_published":"2015-03-01T08:00:00Z","updated_at":"2026-07-24T01:52:45Z","subjects":["Progenitors","Algebra"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://scholarworks.lib.csusb.edu/etd/128","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Hasan, Zahid"]},{"key":"dc:creator","label":"Author","values":["Alharbi, Mashael U"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2015-02-22T08:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Thesis"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Arts in Mathematics"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Progenitors","Algebra"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://scholarworks.lib.csusb.edu/etd/128"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["In this thesis, we have presented our discovery of symmetric presentations of a number of non-abelian simple groups, including the Mathieu group M12. We have given several progenitors, permutation and monomial, including 2*4:(22:3), 2*5:D10, 2*8:((4X2).D4), 3*7:m L2(7), 2*6:(Z3 wr Z2), and 2*24: (2. A5) and their homomorphic images which include 4.(M12:2), the group of automorphisms of M12 and several classical groups. We have given the isomorphism type of each of the group mentioned in the thesis. In each case, a proof of the isomorphism type is provided, either computer-based or by hand. In addition, by hand constructions, using the technique of double coset enumeration, are given for the groups L2(11)X3, L2(11), PGL2(11), S5,(A5XA5):4, A7, and 37:L2(7).
"]},{"key":"dc:title","label":"Title","values":["SYMMETRIC PRESENTATIONS AND RELATED TOPICS"]}]}],"canonical_facts":{"dc:contributor":["Hasan, Zahid"],"dc:creator":["Alharbi, Mashael U"],"dc:date.available":["2015-02-22T08:00:00Z"],"dc:description.abstract":["In this thesis, we have presented our discovery of symmetric presentations of a number of non-abelian simple groups, including the Mathieu group M12. We have given several progenitors, permutation and monomial, including 2*4:(22:3), 2*5:D10, 2*8:((4X2).D4), 3*7:m L2(7), 2*6:(Z3 wr Z2), and 2*24: (2. A5) and their homomorphic images which include 4.(M12:2), the group of automorphisms of M12 and several classical groups. We have given the isomorphism type of each of the group mentioned in the thesis. In each case, a proof of the isomorphism type is provided, either computer-based or by hand. In addition, by hand constructions, using the technique of double coset enumeration, are given for the groups L2(11)X3, L2(11), PGL2(11), S5,(A5XA5):4, A7, and 37:L2(7).
"],"dc:identifier":["https://scholarworks.lib.csusb.edu/etd/128"],"dc:subject":["Progenitors","Algebra"],"dc:title":["SYMMETRIC PRESENTATIONS AND RELATED TOPICS"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Thesis"],"thesis:degree_name":["Master of Arts in Mathematics"]},"updated_at":"2026-07-24T01:52:45Z"}