Anatomy of the Monster: I

New computations in the Monster

“…By character calculations alone, Thompson had shown that the Monster was a quotient of the triangle group ∆(2, 3, 29) = x, y, z | x 2 = y 3 = z 29 = xyz = 1 , and the challenge was to find the minimal value of n such that the Monster is a quotient of ∆(2, 3, n). From Norton's work on maximal subgroups [17] it seemed very likely that this minimal value was 7. However, the probability that a random pair of elements of orders 2 and 3 has product of order 7 is around 10 −8 , so we would need to look at something like 100 million pairs to have a reasonable chance of finding (2, 3, 7)-generators for the Monster.…”

“…A great deal of theoretical work on classifying the maximal subgroups of the Monster has been done in [22,16,17,18], which reduced the problem to classifying conjugacy classes of simple subgroups of just 22 isomorphism types, subject to a variety of other conditions. In her PhD thesis [6] Beth Holmes dealt with 11 of the 22 isomorphism types, namely L 2 (q) for q = 9, 11, 19, 23, 29, 31, 59, 71 and L 3 (4), U 4 (2) and M 11 .…”

Computing in the Monster

“…These are referred to in Norton (1996aNorton ( , 1998a under the name footballs, and defined and discussed under their present name in Norton (1998b). Further information may be obtained in the references cited in that paper.…”

Every in the Monster contains -elements

“…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.…”

“…In [6] some subgroups isomorphic to PSL 2 (13) are described. Notation here and later generally follows the Atlas [2], with one or two modifications following [9].…”