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.
","abstract_html":"<p>In this thesis, we will give our discovery of original symmetric presentations of several important groups. We have investigated permutation and monomial progenitors 2<sup>*8</sup>: (2<sup>3</sup>: 2<sup>2</sup>), 2*9: (3<sup>2</sup>: 2<sup>4</sup>), 2<sup>*10</sup>: (2<sup>4</sup>: (2 × 5)), 5<sup>*4</sup>:<sub>m</sub> (2<sup>3</sup>: 2<sup>2</sup>), 7<sup>*8</sup>:<sub>m</sub> (3<sup>2</sup>: 2<sup>4</sup>), and 3<sup>*5</sup>:<sub>m</sub> (2<sup>4</sup>: (2 × 5)). The finite images of the above progenitors include the Mathieu sporadic group M<sub>12</sub>, the linear groups L<sub>2</sub>(8) and L<sub>2</sub>(13), and the extensions S<sub>6</sub> × 2, 2<sup>8</sup> : .L<sub>2</sub>(8) , and 2<sup>7</sup> : .A<sub>5</sub>. We will show our construction of the four groups S<sub>3</sub> , L<sub>2</sub>(8), L<sub>2</sub>(13), and S<sub>6</sub> × 2 over S<sub>3</sub>, 2<sup>2</sup>, S<sub>3</sub> : 2, and S<sub>5</sub>, 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 L<sub>2</sub>(8) does not satisfy the conditions of Iwasawas Lemma and that the group L<sub>2</sub>(13) is simple by Iwasawas Lemma. We give constructions of M<sub>22</sub> × 2 and M<sub>22</sub> as homomorphic images of the progenitor S<sub>6</sub>.</p>","abstract_has_math":false,"creators":["Marouf, Manal Abdulkarim, Ms."],"institution":null,"degree_name":"Master of Arts in Mathematics","degree_level":"Thesis","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Zahid Hasan"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2015,"date_issued":"2015-09-01T07:00:00Z","date_published":"2015-09-01T07:00:00Z","updated_at":"2026-07-24T01:52:53Z","subjects":["Symple Groups","linear group","extensions Homomorphic Image","Isomorphicm Type","Progenitor","Group","Permutation","Monomial","Character","Classes","Magma","Double Coset Enumeration","Mathieu sporadic group","Physical Sciences and Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://scholarworks.lib.csusb.edu/etd/239","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Zahid Hasan"]},{"key":"dc:creator","label":"Author","values":["Marouf, Manal Abdulkarim, Ms."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2015-08-11T07:00:00Z"]},{"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":["Symple Groups","linear group","extensions Homomorphic Image","Isomorphicm Type","Progenitor","Group","Permutation","Monomial","Character","Classes","Magma","Double Coset Enumeration","Mathieu sporadic group","Physical Sciences and Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://scholarworks.lib.csusb.edu/etd/239"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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.
"]},{"key":"dc:title","label":"Title","values":["Simple Groups and Related Topics"]}]}],"canonical_facts":{"dc:contributor":["Zahid Hasan"],"dc:creator":["Marouf, Manal Abdulkarim, Ms."],"dc:date.available":["2015-08-11T07:00:00Z"],"dc:description.abstract":["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.
"],"dc:identifier":["https://scholarworks.lib.csusb.edu/etd/239"],"dc:subject":["Symple Groups","linear group","extensions Homomorphic Image","Isomorphicm Type","Progenitor","Group","Permutation","Monomial","Character","Classes","Magma","Double Coset Enumeration","Mathieu sporadic group","Physical Sciences and Mathematics"],"dc:title":["Simple Groups and Related Topics"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Thesis"],"thesis:degree_name":["Master of Arts in Mathematics"]},"updated_at":"2026-07-24T01:52:53Z"}