Mathieu Group M24 Research Articles

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:

A new random number generator from permutation groups

We describe a new random number generator, RPGM, which is based on the cryptographic system PGM invented by Magliveras in 1976 and subsequently studied by Magliveras and Surkan [10]. PGM relies on a certain method of machine representation for permutation groups. This method allows for encryption and decryption algorithms based on a space-efficient data structure which is called a logarithmic signature for the group. The efficacy of RPGM is studied by means of an extensive analysis of generated data of 100,000 numbers using the Mathieu groupM 24 in its 5-transitive representation on 24 points.

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

Finite groups in which a Sylow two-subgroup of the centralizer of some involution is of order 16

It is proved that the sectional two-rank of a finite group G having no subgroup of index two is at most four if a Sylow two-subgroup of the centralizer of some involution of G is of order 16. This implies the following assertion: If G is a finite simple group whose order is divisible by 25 and the order of the centralizer of some involution of G is not divisible by 25, then G is isomorphic to the Mathieu group M12 or the Hall-Janko group J2.

Binary self-dual codes of length 24

There are 26 distinct indecomposable self-dual codes of length 24 over GF(2), including unique codes of minimum weights 8 and 6, whose groups are, respectively, the Mathieu group M24 and the maximal subgroup of index 1771 in M^> For each code we give the order of its group, the number of equivalent codes, and its weight distribution.

Primitive groups having transitive subgroups of smaller, prime power degree

The groups in the title are classified, provided they are not too highly transitive. Let G be a primitive permutation group on a finite set S of n points. In 1871, Jordan initiated the study of G under the additional assumption that there is a transitive subgroup H of degree m, where 1 3; the fu l l collineation group o] AG(d,2), where d >= 3; or a Mathieu group M22, Aut(M22), M23, or M24. This result contains several recent theorems found in [2], [3] and [4]. PROOF. G is 2-transitive on S ([7, p. 32]). By 12, Sect. 6], we may assume that G is not 3-transitive. Let B be the complement of the given set of m points, so I B[ = k = n m. Let P be a Sylow subgroup of the pointwise stabilizer G(B) of B, so that P is transitive on S B. By [2, (3.6)], the distinct sets B g, g E G, form a design ~ whose lines have more than two points; moreover, if B is not a line o f ~ , then planes o f ~ are well-defined and G is transitive on the set of planes. Suppose first that B is a line o f ~ . Let O e G be such that B n B ~ is a point x. Since P is transitive on the lines # B on x, the stabilizer P , of B g in P has index t This research was supported in part by NSF Grant GP 37982X. Received January 1, 1974

The Mathieu group M12

Let G be a finite simple non-Abelian group. t is an involution of G, and L=O2 (CG(t)/O · (CG(t))). K the center Z(L) is cyclic and L/Z(L)-PGL (2, q), q odd, then either a Sylow 2-subgroup of G is semidihedral or CG(t)Z2×PGL (2.5) and G is isomorphic to the Mathieu group M12 of degree 12.

On subgroups of 𝑀₂₄. I. Stabilizers of subsets

In this paper we study the orbits of the Mathieu group M 24 {M_{24}} on sets of n points, 1 ≦ n ≦ 12 1 \leqq n \leqq 12 . For n ≧ 6 , M 24 n \geqq 6,{M_{24}} is not transitive on these sets, so we may classify the sets into types corresponding to the orbits of M 24 {M_{24}} and then show how to construct a set of each type from smaller sets. We determine the stabilizer of a set of each type and describe its representation on the 24 points. From the conclusions, the class of subgroups which are maximal among the intransitives of M 24 {M_{24}} can be read off. This work forms the first part of a study which yields, in particular, a complete list of the primitive representations of M 24 {M_{24}} .

A representation of the Mathieu groupM 24 as a collineation group

The Mathieu group M24 can be represented as a collineation group in space of 11 dimensions over the field of two elements. This paper discusses the geometry associated with this representation.

A characterization of the Mathieu group 𝔐₁₂

In this paper we give a characterization of the simple Mathieu group 3J112 of order 95,040. The character table of 9RM2 was computed by Frobenius L6], and from his results it can be immediately seen that there exists an element F in 9I12 of order 8 such that (i) the cyclic subgroup generated by F is self-centralizing, (ii) F is conjugate to its odd powers. Elementary arguments show that there is then a Sylow 2-subgroup $ of 912 Of order 64, such that (i), (ii) hold in $3. We are thus led to a consideration of 2-groups 3 of order 64 in which (i), (ii) hold, and groups ( containing $3 as a Sylow 2-subgroup. Our main result is the following:

Twelve points in PG (5, 3) with 95040 self-transformations

The title of this paper could have been ‘Geometry in five dimensions over GF (3)’ (cf. Edge 1954), or ‘The geometry of the second Mathieu group’, or ‘Duads and synthemes’, or ‘Hexastigms’, or simply ‘Some thoughts on the number 6’. The words actually chosen acknowledge the inspiration of the late H. F. Baker, whose last book (Baker 1946) develops the idea of duads and synthemes in a different direction. The special property of the number 6 that makes the present development possible is the existence of an outer automorphism for the symmetric group of this degree. The consequent group of order 1440 is described abstractly in §1, topologically in §2, and geometrically in §§3 to 7. The kernel of the geometrical discussion is in §5, where the chords of a non-ruled quadric in the finite projective space PG (3, 3) are identified with the edges of a graph having an unusually high degree of regularity (Tutte 1958). It is seen in §4 that the ten points which constitute this quadric can be derived very simply from a ‘hexastigm ’ consisting of six points in PG (4, 3) (cf. Coxeter 1958). The connexion with Edge’s work is described in §6. Then §7 shows that the derivation of the quadric from a hexastigm can be carried out in two distinct ways, sug­gesting the use of a second hexastigm in a different 4-space. It is found in §8 that the consequent configuration of twelve points in PG (5, 3) can be divided into two hexastigms in 66 ways. The whole set of 132 hexastigms forms a geometrical realization of the Steiner system s (5, 6, 12), whose group is known to be the quintuply transitive Mathieu group M 12 , of order 95040. Finally, §9 shows how the same 5-dimensional configuration can be regarded (in 396 ways) as a pair of mutually inscribed simplexes, like Möbius’s mutually inscribed tetrahedra in ordinary space of 3 dimensions.

Mathieu moonshine and the geometry of K3 surfaces

We compare the moonshine observation of Eguchi, Ooguri and Tachikawa relating the Mathieu group M24 and the complex elliptic genus of a K3 surface with the symmetries of geometric structures on K3 surfaces. Two main results are that the complex elliptic genus of a K3 surface is a virtual module for the Mathieu group M24 and also for a certain vertex operator superalgebra V G where G is the holonomy group.

Symmetries of K3 sigma models

It is shown that the supersymmetry-preserving automorphisms of any non-linear σ-model on K3 generate a subgroup of the Conway group Co1. This is the stringy generalisation of the classical theorem, due to Mukai and Kondo, showing that the symplectic automorphisms of any K3 manifold form a subgroup of the Mathieu group M23. The Conway group Co1 contains the Mathieu group M24 (and therefore in particular M23) as a subgroup. We confirm the predictions of the Theorem with three explicit CFT realisations of K3: the T 4 /Z2 orbifold at the self-dual point, and the two Gepner models (2) 4 and (1) 6 . In each case we demonstrate that their symmetries do not form a subgroup of M24, but lie inside Co1 as predicted by our