Symmetric Presentations, Representations, and Related Topics

Author

Adam Manriquez, California State University - San BernardinoFollow

Date of Award

6-2018

Document Type

Thesis

Degree Name

Master of Arts in Mathematics

Department

Mathematics

First Reader/Committee Chair

Hasan, Zahid

Abstract

The purpose of this thesis is to develop original symmetric presentations of finite non-abelian simple groups, particularly the sporadic simple groups. We have found original symmetric presentations for the Janko group J₁, the Mathieu group M₁₂, the Symplectic groups S(3,4) and S(4,5), a Lie type group Suz(8), and the automorphism group of the Unitary group U(3,5) as homomorphic images of the progenitors 2^*60 : (2 x A₅), 2^{*60 :} A₅, 2^*56 : (2³ : 7), and 2^*28 : (PGL(2,7):2), respectively. We have also discovered the groups 2⁴ : A₅, 3⁴ : S₅, PSL(2,31), PSL(2,11), PSL(2,19), PSL(2,41), A₈, 3⁴ : S₅, A₅², 2^• A₅^2, 2 : A₆², PSL(2,49), 2⁸ : A₅, PGL(2,19), PSL(2,71), 2⁴ : A₅, 2⁴ : A₆, PSL(2,7), 3 x PSL(3,4), 2^• PSL(3,4), PSL(3,4), 2^• (M₁₂ : 2), 3⁷:S₇, 3⁵ : S₅, S₆, 2⁵ : S₆, 3⁵ : S₆, 2⁵ : S₅, 2⁴ : S₆, and M₁₂ as homomorphic images of the permutation progenitors ^2*60 : (2 x A₅), 2^*60 : A₅, 2^*21 : (7: 3), 2^*60 : (2 x A₅), 2^*120 : S₅, and ^2*144 : (3² : 2⁴). We have given original proof of the 2^*n Symmetric Presentation Theorem. In addition, we have also provided original proof for the Extension of the Factoring Lemma (involutory and non-involutory progenitors). We have constructed S₅, PSL(2,7), and U(3,5):2 using the technique of double coset enumeration and by way of linear fractional mappings. Furthermore, we have given proofs of isomorphism types for 7 x 2², U(3,5):2, 2^•(M₁₂ : 2), and (4 x 2) :^• 2².

Recommended Citation

Manriquez, Adam, "Symmetric Presentations, Representations, and Related Topics" (2018). Electronic Theses, Projects, and Dissertations. 711.
https://scholarworks.lib.csusb.edu/etd/711