Simple Groups and Related Topics

In this thesis, we will give our discovery of original symmetric presentations of several important groups. We have investigated permutation and monomial progenitors 2*8: (23: 22), 2*9: (32: 24), 2*10: (24: (2 × 5)), 5*4:m (23: 22), 7*8:m (32: 24), and 3*5:m (24: (2 × 5)). The finite images of the above progenitors include the Mathieu sporadic group M12, the linear groups L2(8) and L2(13), and the extensions S6 × 2, 28 : .L2(8) , and 27 : .A5. We will show our construction of the four groups S3 , L2(8), L2(13), and S6 × 2 over S3, 22, S3 : 2, and S5, by using the technique of double coset enumeration. We will also provide isomorphism types all of the groups that have appeared as finite homomorphic images. We will show that the group L2(8) does not satisfy the conditions of Iwasawas Lemma and that the group L2(13) is simple by Iwasawas Lemma. We give constructions of M22 × 2 and M22 as homomorphic images of the progenitor S6.