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29

Oct

2024

Oct

2024

**1 Background**

In 1873 the French mathematician Emil Mathieu published a paper in which he ’glued’ together copies of the projective special linear group L_{2}(23) acting on the 24-point projective line P_{1}(23) to produce a new group with remarkable properties. Although enormous in comparison to L_{2}(23), the new group was a tiny subgroup of A_{24}, the alternating group on 24 letters. Moreover, it was quintuply transitive; that is to say that, given two ordered quintuples of distinct letters from the 24, *{x*_{1}*, x*_{2}*, . . . , x*_{5}*} *and *{y*_{1}*, y*_{2}*, . . . , y*_{5}*} *say, the group possessed an element mapping *x _{i} → y_{i} *for

For decades the situation remained like this and there seems to have been a tacit acceptance that the five Mathieu groups were the only *sporadic *finite simple groups. Then, out of the blue in 1965, Zvonimir Janko produced a new finite simple group J_{1}, thus launching the most exciting period in modern group theory as mathematicians around the world started to search for other new sporadic simple groups.

**2 The Leech lattice and the Conway group **

M_{24} possesses a class of involutions of cycle shape 1^{8}*.*2^{8}. The 8-element fixed point subsets of these involutions are known as the *octads*; quintuple tran- sitivity implies that every 5-point subset must belong to the same number of octads. In fact, this number is one, and the octads form a *Steiner system *S(5*, *8*, *24). If we let each octad correspond to a 24-dimensional vector over Z_{2}, the field of order 2, having 1 in the positions of the octad and 0 else where, then the octads span a 12-dimensional subspace known as the *binary Golay code C *.

In 1964, John Leech used M_{24} and *C *to construct a new 24-dimensional lattice which had important *sphere-packing *properties. It had the sym- metries of M_{24} and sign changes on octads built in, but Leech had ample evidence to suggest that its full group of automorphisms was much larger. John McKay persuaded John Conway to look into it and, in a beautifully elegant piece of work Conway constructed the group *·*O which now bears his name. Factoring *·*O by its centre of order 2, we obtain Co_{1}, the largest Conway group, a new sporadic simple group. Co_{1} contains two further new sporadic simple groups Co_{2} and Co_{3} which are the stabilizers of certain vec- tors in the lattice. Many of the other new groups that had been discovered during this flurry of activity are also involved in Co_{1}.

**3 The Miracle Octad Generator**

In 1968 Conway took me on as his research student and, soon after, set me to work on his recently discovered groups. It became apparent that an intimate understanding of the underlying Steiner system was imperative. At that time people were reliant on the list of all 759 octads printed in a paper by John Todd, but this was cumbersome and unenlightening. I resolved to to produce a device which would enable one to immediately recognise the octads and, indeed, given any five points to construct the unique octad containing them. I displayed the 24 points as a 4 *×*6 array and, using Todd’s list, began to knock it into shape. One evening, playing around with it in a pub called *The Cricketers’ Arms*, I realised that the system had a 6-fold symmetry which meant that I could encapsulate the whole thing in just 35 small pictures. The result is the Miracle Octad Generator or MOG, see Figure 1.**4 Early uses of the MOG**

The availability of the MOG facilitated detailed work with M_{24} and *·*O: proofs of uniqueness of the S(5*, *8*, *24) and of the completeness of the list of maximal subgroups of M_{24}; a minimal test to see whether a permutation on 24 letters lies in the favoured copy of M_{24}, and so on. It provides a convenient and widely used notation for elements of M_{24} and for vectors in the Leech lattice. It is not only useful in working with sporadic groups contained in *·*O, but also with the larger Fischer groups and the largest Janko group J_{4} which involve Mathieu groups.**5 Subsequent developments**

**5.1 Symmetric generation of groups**

Familiarity with M_{24} led to my realising that the group could be generated by a highly symmetric set of 7 involutions which can be written down directly from the action of L_{2}(7) on 24 letters. This action can be realised combinatorially, or geometrically acting on the 24 faces of the Klein map. This in turn led me to the concept of *symmetric generation of groups *which has provided spontaneous constructions of many of the other sporadic groups. Among my favourites are the delightful Higman-Sims group and J_{4}, mentioned above.

**5.2 The Thompson chain of groups**

This in turn led to my considering a configuration in which the edges of a complete directed graph on *n *vertices are interpreted as elements of order 7 in some group; if *t _{ij} *is the element corresponding to the edge

**6 Interaction with computation**

All of the above are explained in detail in the book *The Art of working with the Group *M_{24}, CUP (2025). There is a constant interplay between compu- tation using the computer package MAGMA and theoretical techniques. It is an intention of the book to emphasise how the two approaches cross-fertilize in this gorgeous combinatorial and algebraic environment.

Title: The Art of Working with the Mathieu Group M24

Part of Cambridge Tracts in Mathematics

Author: Robert T. Curtis

ISBN: 9781009405676

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