A New Maximal Subgroup of the Monster

Abstract

We use our computer construction of the Monster sporadic simple group to find a new maximal subgroup PGL 2 (29). In particular, we prove containment of L 2 (29)

A classification of subgroups of the Monster isomorphic to S4 and an application

“…Doing this for the different classes of A 4 gives possible S 4 centralisers in M. Together with the (2,3,4) structure constants, this is enough to classify S 4 's for some class fusions.…”

“…These are classified using the 196882 dimensional computer representation over GF (3) of [2] as described in Section 4.4.…”

“…Further techniques for using the construction were developed in [3] and [4]. We summarise them below.…”

Every in the Monster contains -elements

“…Co 1 . However, a method was given in [3] for 'changing post', specifically, finding a word in the generators of the Monster which conjugates any given 2B-element in 2 1+24. Co 1 to the central involution.…”

“…Much work has however been done on this problem over the years (see for example [3,[5][6][7][8]10]), but a few obstinate cases remain. One of these is the problem of classifying subgroups isomorphic to PSL 2 (13), whose normalizers might be maximal.…”