Symmetric Presentations of Finite Groups

Abstract

It is often the case that a progenitor, P = m* n : N, factored by a subgroup generated by one or more relators, M = {iriwi, , 7TfcWfc), gives a finite group F, particularly, a classical group, simple group, or a sporadic group. In such instances, the presentation of the factor group, G = P/M = {x, y, t), is also a symmetric presentation of the finite group F. Symmetric presentations of groups allow us to represent, and ma­ nipulate, group elements in a manner that is typically more convenient than conventional techniques; in this sense, symmetric presentations are particularly useful in the study of large finite groups. In this thesis, we first construct, by manual double coset enumeration, the groups A5, S5, Sq, S7, and S7 x 3 as finite homomorphic images of the progenitors 2* 3 : S3, 2* 4 : A4, 2* 5 : A5, 3* 5 : S5, and 3* 5 : S5, respectively. We also demonstrate that their respective symmetric presentations enable us to represent, and manipulate, their group elements in a convenient (symmetric) fashion as well as to obtain, in most cases, useful permutation representations for their group elements. We devote the majority of our efforts to the construction, and manipulation, of M12: 2, or Aut(Mi2), the outer automorphism group of the Mathieu group M12. In particular, we construct, by the technique of manual double coset enumeration over S4, the group Aut(Mi2) as a finite homomorphic image of the progenitor 3* 4 : S4. By way of this construction, we show that Aut(Afi2) is isomorphic to 3* 4 : S4 factored by two relations and we conclude that the symmetric presentation (x, y,t | x4 = y2 = (yx)3 = t3 = [t,y] = [i33,?/] = (yxt)10 = ((x2y)2t)5 = e) defines the group Aut(Mi2). Finally, we demonstrate that this symmetric presentation enables us to express and manipulate every element of Aut(Mi2) either as a symmetric representation of the form 7rw, where tt is a permutation of S4 on 4 letters and w is a word of concatenated generators of length at most eight, or as a permutation representation on 7920 letters.