The Mathieu Group M 12 and Conway ’ s M 13 - Game Senior Honors

Introduction Although the Mathieu groups are probably best known today as the first instances of the sporadic simple groups, Mathieu's discovery of them arose rather from his search for highly transitive permutation groups; only later were the Mathieu groups shown to be simple. In modern mathematics, the Mathieu groups are studied chiefly in conjunction with sphere packing and error-correcting codes. Conway [3] has recently proposed an unusual method of constructing the Mathieu group M 12 , which has a natural extension to a larger quasigroup, heretofore unknown, named by Conway " M 13 ". Con-way's original article leaves the investigation of this construction as an open problem. We begin from a classical point of view, describing the Mathieu groups in terms of their most salient properties, multiple transitivity and simplicity. With this material in the background, we move to a discussion of Conway's construction of the group M 12 and the quasigroup M 13 , as set forth in Con-way's original article and a subsequent paper of Elkies [6]. In particular, we examine a metric on M 13 , induced naturally by the Conway construction, and determine the extent to which M 13 is sextuply transitive. Finally, we discuss an extension of the Conway construction, known to Conway and explored further by Elkies, inducing signed extensions 2M 13 and 2M 12. This thesis consists in part of a summary of previous work on the Mathieu groups, and in part of original work by the author. In particular, the sections dealing with the metrization of M 13 , sextuple transitivity, and antipodes in 2M 13 , culminating in Theorems IV.1, IV.4, IV.10, and IV.15, are my own research, carried out under the supervision of Professor Noam Elkies. This research was facilitated by a computer program that I designed to implement the Conway construction. Thanks are due, first and foremost, to my advisor, Professor Noam D. Elkies, for his devoted attention to every aspect of this work. I would also like to acknowledge Professor M. Krishnamoorthy for providing me with Conway's seminal article on M 13 , and finally Professor John H. Conway himself, without whose insight much of this thesis would not exist.