M24 and the MOG
( A. Taormina)

Description

The Mathieu group M24 is one of the 26 exceptional groups (the so-called sporadic groups, of which the Monster group is a famous example) in the classification of finite simple groups. A simple group does not have normal subgroups, apart from itself and the identity.

In many respects, M24 is remarkable. From purely numerical curiosities to deep connections with the description of topological invariants of two complex dimensional hyperkahler manifolds (K3 manifolds), this group also plays a pivotal role in the construction of the Leech lattice, discovered in 1965 in connection with the packing of spheres in 24-dimensional space.

Recently, M24 has come under the intense scrutiny of Number Theorists and Theoretical Particle Physicists working in String Theory, because of a remarkable object, called a `mock modular form', that encodes, in its Fourier expansion, the dimensions of irreducible representations of M24 and of their tensor products. This is reminiscent of a similar remarkable and very deep phenomenon related to the `Monster Group', dubbed the `Monster Moonshine'. The big question is: `Is there an M24 Moonshine?'

M24 is a subgroup of order 244823040 of the alternate group A24, the group of even permutations of 24 elements. Given its size, one must device clever tools to fully appreciate its structure. One tool is called the Miracle Octad Generator (or MOG), invented by Rob Curtis in the seventies while working with John Conway, his PhD supervisor.

The project will explore the relation between M24, the Steiner system S(5,8,24) and the extended binary Golay code, through the use of the MOG.

Prerequisites

Resources

-The book `Sphere packings, lattices and groups' by J.H. Conway and N.J.A. Sloane, Springer Verlag, ISBN 0-387-96617-X is a good source of information on M24.

-The original paper `A new combinatorial approach to M24', by R.T. Curtis, Math.Proc.Camb.Phil.Soc. (1976), 79,25 is worth looking at too.

-For general information, consult the wikipedia entry for Mathieu groups and sporadic groups.

-If you like geometric visualisations, visit http://homepages.wmich.edu/~drichter/mathieu.htm on how to make the Mathieu group M24.

-You can brush up your group theory knowledge by reading `An introduction to the theory of groups' by Joseph Rotman, Springer ISBN 0-387-94285-8, and the first few chapters of `Finite group theory' by M. Aschbacher, Cambridge studies in advanced mathematics, ISBN 0-521-45826-9.