Symmetric Presentation of Finite Groups, and Related Topics

Author

Marina Michelle DuchesneFollow

Date of Award

5-2021

Document Type

Thesis

Degree Name

Master of Arts in Mathematics

Department

Mathematics

First Reader/Committee Chair

Hasan, Zahid

Abstract

We have discovered original symmetric presentations for several finite groups, including 2²:^.(2⁴:(2^.S₃)), M₁₁, 3:(PSL(3,3):2), S₈, and 2^.M₁₂. We have found homomorphic images of several progenitors, including 2^*18:((6x2):6), 2^*24:(2^.S₄), 2^*105:A₇, 3^*3:_m(2³:3), 7^*8:_m(PSL(2,7):2), 3^*4:_m(4²:2²), 7^*5:(2xA₅), and 5^*6:_mS₅. We have provided the isomorphism type of all of the finite images that we have discovered. We have found 2²:^.(2⁴:(2^.S₃)), M₁₁, 3:(PSL(3,3):2), S₈, and 2^.M₁₂ as homomorphic images of the progenitor 2^*24:(2^.S₄). We have constructed, through our technique of double coset enumeration, $2^2:^{\bullet}(2^4:(2^{\bullet}S_3))=\frac{2^{*24}:(2^{\bullet}S_4)}{(x*y*t)^2,(x*y*t^{y*x})^4}$, $M_{11}=\frac{2^{*24}:(2^{\bullet}S_4)}{(x*y*x^{-1}*t^x)^2,(x*y*x^{-1}*t^{y^x})^5,(x*y*t^{y*x})^5}$, $3:(PSL(3,3):2)=\frac{2^{*24}:(2^{\bullet}S_4)}{(x*y*x^{-1}*t^{y*x})^3,(x*y*t)^2,(x*y*t^{y*x})^6}$, $S_8=\frac{2^{*24}:(2^{\bullet}S_4)}{(x*y*x^{-1}*t^{y*x})^6,(x*y*x^{-1}*t^{y^x})^6,(x*y*t)^2,(x*y*t^{y*x})^7}$, and $2^{\bullet}M_{12}=\frac{2^{*24}:(2^{\bullet}S_4)}{(x*y*x^{-1}*t^{y*x})^5,(x*y*x^{-1}*t^{y^x})^3,(x*y*t)^3}$.

Recommended Citation

Duchesne, Marina Michelle, "Symmetric Presentation of Finite Groups, and Related Topics" (2021). Electronic Theses, Projects, and Dissertations. 1191.
https://scholarworks.lib.csusb.edu/etd/1191