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.
","abstract_html":"<p>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<sup>*14</sup> : <em>D</em><sub>28</sub>, 2<sup><em>∗</em>9</sup> : 3<sup><em>•</em></sup>(3<sup>2</sup>), 3<sup><em>∗</em>9</sup> : 3<sup><em>•</em></sup>(3<sup>2</sup>), 2<sup><em>∗</em>21</sup> : (7<em>X</em>3) : 2 as well as monomial progenitors, including 7<sup><em>∗</em>5</sup> :<sub><em>m </em></sub><em>A</em><sub>5</sub>, 3<sup><em>∗</em>5</sup> :<em><sub>m</sub> </em><em>S</em><sub>5</sub>. We have included their homomorphic images which include the Mathieu group <em>M</em><sub>12</sub>, 2<sup><em>•</em></sup><em>J</em><sub>2</sub>, 2<em>X</em><em>S</em>(4<em>,</em><em> </em>5), as well as, many <em>P</em><em>GL</em><em>′</em><em>s</em>, <em>P</em><em>S</em><em>L</em><em>′</em><em>s </em>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, 2<sup>5</sup> : <em>S</em><sub>5</sub> over 2<sup><em>∗</em>5</sup> : <em>S</em><sub>5</sub>, <em>P</em><em>S</em><em>L</em>(2<em>, </em>8) over 2<sup><em>∗</em>7</sup> : <em>D</em><sub>14</sub>, <em>M</em><sub>12 </sub>over a maximal subgroup, 2<em>X</em><em>S</em><sub>5</sub>. We have developed a lemma using relations to factor permutation progenitors of the form <em>m</em><sup><em>∗</em></sup><em>n</em><em> </em>: <em>N </em>to give an isomorphism of <em>m</em><sup><em>n</em></sup><em> </em>: <em>N </em>. 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 <em>P</em><em>S</em><em>L</em>(2<em>, </em>8) and <em>M</em><sub>12</sub> to be simple groups.</p>","abstract_has_math":false,"creators":["Grindstaff, Dustin J"],"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":"2015-06-01T07:00:00Z","date_published":"2015-06-01T07:00:00Z","updated_at":"2026-07-24T01:52:45Z","subjects":["symmetric presentation","simple group","progenitor","MAGMA","double coset enumeration","homomorphic image","Algebra"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://scholarworks.lib.csusb.edu/etd/202","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":["Grindstaff, Dustin J"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2015-05-20T07:00:00Z"]},{"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":["symmetric presentation","simple group","progenitor","MAGMA","double coset enumeration","homomorphic image","Algebra"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://scholarworks.lib.csusb.edu/etd/202"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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.
"]},{"key":"dc:title","label":"Title","values":["Symmetric Presentations and Generation"]}]}],"canonical_facts":{"dc:contributor":["Hasan, Zahid"],"dc:creator":["Grindstaff, Dustin J"],"dc:date.available":["2015-05-20T07:00:00Z"],"dc:description.abstract":["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.
"],"dc:identifier":["https://scholarworks.lib.csusb.edu/etd/202"],"dc:subject":["symmetric presentation","simple group","progenitor","MAGMA","double coset enumeration","homomorphic image","Algebra"],"dc:title":["Symmetric Presentations and Generation"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Thesis"],"thesis:degree_name":["Master of Arts in Mathematics"]},"updated_at":"2026-07-24T01:52:45Z"}