Symmetric Presentations and Generation

<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^*14 : <em>D</em>₂₈, 2^{<em>∗</em>9} : 3^<em>•</em>(3²), 3^{<em>∗</em>9} : 3^<em>•</em>(3²), 2^{<em>∗</em>21} : (7<em>X</em>3) : 2 as well as monomial progenitors, including 7^{<em>∗</em>5} :_{<em>m </em>}<em>A</em>₅, 3^{<em>∗</em>5} :<em>_m </em><em>S</em>₅. We have included their homomorphic images which include the Mathieu group <em>M</em>₁₂, 2^<em>•</em><em>J</em>₂, 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, 2⁵ : <em>S</em>₅ over 2^{<em>∗</em>5} : <em>S</em>₅, <em>P</em><em>S</em><em>L</em>(2<em>, </em>8) over 2^{<em>∗</em>7} : <em>D</em>₁₄, <em>M</em>₁₂over a maximal subgroup, 2<em>X</em><em>S</em>₅. We have developed a lemma using relations to factor permutation progenitors of the form <em>m</em>^<em>∗</em><em>n</em><em> </em>: <em>N </em>to give an isomorphism of <em>m</em>^<em>n</em><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>₁₂ to be simple groups.</p>

Name thesis:degree_name

Master of Arts in Mathematics

Level thesis:degree_level

Thesis

Discipline thesis:degree_discipline

Mathematics

Year dc:date.available

2015

Author dc:creator

• Grindstaff, Dustin J

Contributors dc:contributor

• Hasan, Zahid

• symmetric presentation
• simple group
• progenitor
• MAGMA
• double coset enumeration
• homomorphic image
• Algebra

Repository record dc:identifier

https://scholarworks.lib.csusb.edu/etd/202

OAI identifier oai:identifier

oai:scholarworks.lib.csusb.edu:etd-1202