Group M24 Research Articles

The 3-local cohomology of the Mathieu group M24

In this paper we calculate the localisation at the prime 3 of the integral cohomology ring of the Mathieu group M24, together with its mod-3 cohomology ring. The main results areTheorem 1. The ring H*(M24, Z)(3)is the commutative graded Z(3)-algebra with generatorsand relations v2 = 0 and βθ = 0. The Chern classes of the Todd representation in GL11F2

Lacunarity of Dedekind η-products

The Dedekind η-function is defined bywhere τ lies in the upper half plane ℋ = {tau;|Im(τ) > 0}, and x = e2πiτ. It is a modular form of weight ½ with a multiplier system. We define an η-product to be a function f (τ) of the formwhere rδ ε ℤ. This is a modular form of weight with a multiplier system. The Fourier coefficients of η-products are related to many well-known number-theoretic functions, including partition functions and quadratic form representation numbers. They also arise from representations of the “monster” group [3] and the Mathieu group M24 [13]. The multiplicative structure of these Fourier coefficients has been extensively studied. Recent papers include [1], [4], [5] and [6]. Here we study the connections between the density of the non-zero Fourier coefficients of f(τ) and the representability of f(τ) as a linear combination of Hecke character forms (defined in Section 4 below). We first make the following definition.

A conjectured analogue of Dedekind’s eta function for $K3$ surfaces

Ab stract. A fundamental formula in the study of elliptic functions is the product formula for Dedekind’s eta function or, equivalently, for the holomorphic cusp form on the upper half plane h which is of weight 12 with respect to the action by PSL (2, Z). A related formula expresses the determinant of the Laplacian which acts on the space of smooth functions on an elliptic curve with a period of the elliptic curve and the Dedekind eta function. In [JT 94a], we constructed a holomorphic function on the moduli space of marked, polarized, algebraic K3 surfaces of fixed degree using determinants of Laplacians. The aim of this article is to state a conjecture which expresses a product formula for this holomorphic form. In addition, we will present speculative relations with the representation theory of the Mathieu group M24, as well as state many other problems currently under investigation.

On the uniqueness of a regular thin near octagon on 288 vertices (or the semibiplane belonging to the Mathieu group M12)

We show that there are unique distance-regular graphs with intersection arrays {6, 5, 4, 1; 1, 2, 5, 6} and {12, 11, 10, 7; 1, 2, 5, 12}, and that no distance-regular graph with intersection array {10, 9, 8, 5; 1, 2, 5, 10} exists.

Symmetric Presentations II: The Janko Group J 1

In our paper [6] we exploited parallel behaviour of the Mathieu groups M12 and M24 which we described fully in [4, 5], to obtain presentations of those groups given implicitly in terms of certain highly symmetric generating sets. We made precise the concepts of symmetric generating set, symmetric presentation and primitive presentation, and indicated how these ideas can be generalized to other groups. In the present work, which is a continuation of [6], we turn the process around and seek groups which possess generating sets with certain specified symmetries. We shall find that the most straightforward ways of generalizing [6] immediately give rise to the two smallest Janko groups J1 and J2 Moreover the permutation actions of minimal degree of these two groups, on 266 and 100 letters respectively, emerge naturally and we are readily able to write down our symmetric generators in these representations. All assertions made can be easily checked on a computer, and we are again heavily indebted to the CAYLEY package. That said the final developments of J1 and J2 given here, and in the sequel to this paper, are entirely by hand and are complete in all detail.

Representations of M12

We introduce two families of lattices for the symmetric group and use them to build rank 10 and rank 45 lattices for the groups 2M122 and M12, respectively.

A Geometry for M24

We construct an irreducible 45-dimensional representation of the sporadic simple group M24.

A geometric characterization of the groups M12, He and Ru

A geometric characterization of the groups M12, He and Ru

The Symmetric Genus of the Mathieu Groups

The (symmetric) genus of a finite group may be defined as the smallest genus of those closed orientable surfaces on which G acts faithfully as a group of automorphisms. In this paper the genus of each of the five Mathieu groups M11, M12, M22, M23 and M24 is determined, with the help of some computer calculations and a little-known of Ree on permutations.

The geometry and cohomology of the Mathieu group M12

It is usually very complex to calculate the cohomology of a finite group G. To date the most significant results have been on the symmetric and alternating groups ([N], [M– M], [Ma], [Mu], [A–M–M]); the general linear groups over a finite field [Q1]; and the extra–special p–groups [Q2]. In the list above only the alternating groups and the GLn(F2) are simple. Indeed, among the simple groups only a very few, aside from those above have been understood. It is probable that away from the characteristic, p, H(PSLn(Fpn );Z/q) is available, but even this is not obvious, especially for q = 2. And certainly we currently have no information on most of the families of groups of Lie type, for example, G2(Fq ), E6(Fq ), E7(Fq ), E8(Fq ), or any of the sporadic groups but M11 and J1. About 15 years ago Quillen introduced very powerful techniques that allowed for a much deeper understanding of group cohomology. However, it is also apparent that successful calculations over Z/p have only occurred when the group has a well–behaved lattice of p–elementary abelian subgroups. Formidable combinatorial problems arise in the general situation, as well as necessarily complicated multiplicative relations. For example, the general calculation of H(GLn(Fp);Z/p) remains quite inaccessible, and there has been only marginal progress since Quillen’s landmark results.

A character-theory-free characterization of the Mathieu group M12

AbstractThe known characterization of the Mathieu group M12 by the structure of the centralizer of a 2-central involution is based on the application of the theory of exceptional characters and uses in addition a block theoretic result which asserts that a simple group of order |M12| is isomorphic to M12. The details of the proof of the latter result had never been published. We show here that M12 can be handled in a completely elementary and group theoretical way.

On a system of elliptic modular forms attached to the large Mathieu group

This paper is a continuation of two previous papers of the author. In the first [4] we discussed a Thompson series associated with the group M24 in which each of the modular forms ηg(τ) attached to elements g ∈ M24 are primitive cusp-forms.

Geometric interpretations of the ‘natural’ generators of the Mathieu groups

AbstractIn [1] we used the alternating group A5 to produce a set of five permutations of order three acting on 12 letters; this set was normalized by A5 and generated the Mathieu group M12. We analogously used the linear group PSL2(7) to produce a set of seven involutions acting on 24 letters which was normalized by our PSL2(7) and generated M24. In this paper we describe the combinatorial and geometric interpretations of our generators, and aim to justify our use of the word ‘natural’. The generators for M12 are defined both as permutations of twelve 5-cycles in a conjugacy class of A5, and as permutations of the faces of a dodecahedron. The outer automorphism of M12 emerges as a bonus. The generators of M24 are defined analogously both as permutations of twenty-four 7-cycles in the action of PSL2(7) on seven letters, and as permutations of the 24 faces of a certain cubic graph drawn on a surface of genus three. In addition we mention the connection between the latter construction and the binary Golay code.

Further elementary techniques using the miracle octad generator

In this paper we describe various techniques, some of which are already used by devotees of the art, which relate certain maximal subgroups of the Mathieu group M24, as seen in the MOG, to matrix groups over finite fields. We hope to bring out the wealth of algebraic structure underlying the device and to enable the reader to move freely between these matrices and permutations. Perhaps the MOG was mis-named as simply an “octad generator”; in this paper we intend to show that it is in reality a natural diagram of the binary Golay code.

The subgroup structure of the Mathieu group 𝑀₁₂

We present a full description of the lattice of subgroups of the Mathieu group M 12 {M_{12}} .

From Error-Correcting Codes through Sphere Packings to Simple Groups.

1. The origin of error-correcting codes an introduction to coding the work of Hamming the Hamming-Holbrook patent the Hamming codes are linear the work of Golay the priority controversy 2. From coding to sphere packing an introduction to sphere packing the Leech connection the origin of Leech's first packing in E24 the matrix for Leech's first packing the Leech lattice 3. From sphere packing to new simple groups is there an interesting group in Leech's lattice? the hard sell of a simple group twelve hours on Saturday, six on Wednesday the structure of 0 new simple groups Appendix 1. Densest known sphere packings Appendix 2. Further properties of the (12,24) Golay code and the related Steiner system s (5,8,24) Appendix 3. A calculation of the number of spheres with centers in LAMBDA2 Appendix 4. The Mathieu group M24 and the order of M22 Appendix 5. The proof of lemma 3.3 Appendix 6. The sporadic simple groups Bibliography Index.

The theta functions of sublattices of the Leech lattice

Let Λ be the Leech lattice which is an even unimodular lattice with no vectors of squared length 2 in 24-dimensional Euclidean space R24. Then the Mathieu Group M24 is a subgroup of the automorphism group .0 of Λ and the action on Λ of M24 induces a natural permutation representation of M24 on an orthogonal basis For , let Λm be the sublattice of vectors invariant under m:

Mathieu group M24 and modular forms

In [6], Mason reported some connections between sporadic simple group M24 and certain cusp forms which appear in the ‘denominator’ of Thompson series assigned to Fisher-Griess’s group F1. In this paper, we discuss the ‘numerator’ of these Thompson series.

The projective characters of the Mathieu group M12 and of its automorphism group

In (1), Burgoyne and Fong have shown that the Schur multiplier of the Mathieu group M12 is of order 2. It is shown in Theorem 2·4 that the Schur multiplier of Aut M12, the automorphism group of M12, is also of order 2. It is therefore possible to choose a complex 2-cocycle α of Aut M12, taking only the values 1 and − 1, such that the cohomology class of α is of order 2 and the cohomology class of the restriction of α to M12 is of order 2. The characters of the irreducible α-projective representations of Aut M12 are calculated in § 2.

Algorithms for enumerating elements of mathieu groups M11 and M12, and error-correction in codes equivalent to these groups

Algorithms for enumerating elements of mathieu groups M11 and M12, and error-correction in codes equivalent to these groups