# Symmetric Presentation of Finite Groups, and Related Topics

5-2021

Thesis

## Degree Name

Master of Arts in Mathematics

Mathematics

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 $2^{\bullet}M_{12}=\frac{2^{*24}:(2^{\bullet}S_4)}{(x*y*x^{-1}*t^{y*x})^5,(x*y*x^{-1}*t^{y^x})^3,(x*y*t)^3}$.

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