Conway group

Thomas Thompson (1983) relates how, in about 1964, John Leech investigated close packings of spheres in Euclidean spaces of large dimension. One of Leech's discoveries was a lattice packing in 24-space, based on what came to be called the Leech lattice Λ. He wondered whether his lattice's symmetry group contained an interesting simple group, but felt he needed the help of someone better acquainted with group theory. He had to do much asking around because the mathematicians were pre-occupied with agendas of their own. John Conway agreed to look at the problem. John G. Thompson said he would be interested if he were given the order of the group. Conway expected to spend months or years on the problem, but found results in just a few sessions.

Witt (1998, page 329) stated that he found the Leech lattice in 1940 and hinted that he calculated the order of its automorphism group Co₀.

Conway started his investigation of Co₀ with a subgroup he called N, a holomorph of the (extended) binary Golay code (as diagonal matrices with 1 or −1 as diagonal elements) by the Mathieu group M₂₄ (as permutation matrices). N ≈ 2¹²:M₂₄.

A standard representation, used throughout this article, of the binary Golay code arranges the 24 co-ordinates so that 6 consecutive blocks (tetrads) of 4 constitute a sextet.

The matrices of Co₀ are orthogonal; i. e., they leave the inner product invariant. The inverse is the transpose. Co₀ has no matrices of determinant −1.

The Leech lattice can easily be defined as the Z-module generated by the set Λ₂ of all vectors of type 2, consisting of

(4, 4, 0²²)

(2⁸, 0¹⁶)

(−3, 1²³)

and their images under N. Λ₂ under N falls into 3 orbits of sizes 1104, 97152, and 98304. Then |Λ₂| = 196,560 = 2⁴⋅3³⋅5⋅7⋅13. Conway strongly suspected that Co₀ was transitive on Λ₂, and indeed he found a new matrix, not monomial and not an integer matrix.

Let η be the 4-by-4 matrix

${\displaystyle {\frac {1}{2}}{\begin{pmatrix}1&-1&-1&-1\\-1&1&-1&-1\\-1&-1&1&-1\\-1&-1&-1&1\end{pmatrix}}}$

Now let ζ be a block sum of 6 matrices: odd numbers each of η and −η.[1][2] ζ is a symmetric and orthogonal matrix, thus an involution. Some experimenting shows that it interchanges vectors between different orbits of N.

To compute |Co₀| it is best to consider Λ₄, the set of vectors of type 4. Any type 4 vector is one of exactly 48 type 4 vectors congruent to each other modulo 2Λ, falling into 24 orthogonal pairs {v, –v}. A set of 48 such vectors is called a frame or cross. N has as an orbit a standard frame of 48 vectors of form (±8, 0²³). The subgroup fixing a given frame is a conjugate of N. The group 2¹², isomorphic to the Golay code, acts as sign changes on vectors of the frame, while M₂₄ permutes the 24 pairs of the frame. Co₀ can be shown to be transitive on Λ₄. Conway multiplied the order 2¹²|M₂₄| of N by the number of frames, the latter being equal to the quotient |Λ₄|/48 = 8,252,375 = 3⁶⋅5³⋅7⋅13. That product is the order of any subgroup of Co₀ that properly contains N; hence N is a maximal subgroup of Co₀ and contains 2-Sylow subgroups of Co₀. N also is the subgroup in Co₀ of all matrices with integer components.

Since Λ includes vectors of the shape (±8, 0²³), Co₀ consists of rational matrices whose denominators are all divisors of 8.

The smallest non-trivial representation of Co₀ over any field is the 24-dimensional one coming from the Leech lattice, and this is faithful over fields of characteristic other than 2.

Any involution in Co₀ can be shown to be conjugate to an element of the Golay code. Co₀ has 4 conjugacy classes of involutions.

A permutation matrix of shape 2¹² can be shown to be conjugate to a dodecad. Its centralizer has the form 2¹²:M₁₂ and has conjugates inside the monomial subgroup. Any matrix in this conjugacy class has trace 0.

A permutation matrix of shape 2⁸1⁸ can be shown to be conjugate to an octad; it has trace 8. This and its negative (trace −8) have a common centralizer of the form (2¹⁺⁸×2).O₈⁺(2), a subgroup maximal in Co₀.

Conway and Thompson found that four recently discovered sporadic simple groups, described in conference proceedings (Brauer & Sah 1969), were isomorphic to subgroups or quotients of subgroups of Co₀.

Conway himself employed a notation for stabilizers of points and subspaces where he prefixed a dot. Exceptional were .0 and .1, being Co₀ and Co₁. For integer n ≥ 2 let .n denote the stabilizer of a point of type n (see above) in the Leech lattice.

Conway then named stabilizers of planes defined by triangles having the origin as a vertex. Let .hkl be the pointwise stabilizer of a triangle with edges (differences of vertices) of types h, k and l. The triangle is commonly called an h-k-l triangle. In the simplest cases Co₀ is transitive on the points or triangles in question and stabilizer groups are defined up to conjugacy.

Conway identified .322 with the McLaughlin group McL (order 898,128,000) and .332 with the Higman–Sims group HS (order 44,352,000); both of these had recently been discovered.

Here is a table[3][4] of some sublattice groups:

Two sporadic subgroups can be defined as quotients of stabilizers of structures on the Leech lattice. Identifying R²⁴ with C¹² and Λ with

${\displaystyle \mathbf {Z} \left[e^{{\frac {2}{3}}\pi i}\right]^{12},}$

the resulting automorphism group (i.e., the group of Leech lattice automorphisms preserving the complex structure) when divided by the six-element group of complex scalar matrices, gives the Suzuki group Suz (order 448,345,497,600). This group was discovered by Michio Suzuki in 1968.

A similar construction gives the Hall–Janko group J₂ (order 604,800) as the quotient of the group of quaternionic automorphisms of Λ by the group ±1 of scalars.

The seven simple groups described above comprise what Robert Griess calls the second generation of the Happy Family, which consists of the 20 sporadic simple groups found within the Monster group. Several of the seven groups contain at least some of the five Mathieu groups, which comprise the first generation.

Co₀ has 4 conjugacy classes of elements of order 3. In M₂₄ an element of shape 3⁸ generates a group normal in a copy of S₃, which commutes with a simple subgroup of order 168. A direct product PSL(2,7) × S₃ in M₂₄ permutes the octads of a trio and permutes 14 dodecad diagonal matrices in the monomial subgroup. In Co₀ this monomial normalizer 2⁴:PSL(2,7) × S₃ is expanded to a maximal subgroup of the form 2.A₉ × S₃, where 2.A₉ is the double cover of the alternating group A₉.

John Thompson pointed out it would be fruitful to investigate the normalizers of smaller subgroups of the form 2.A_n (Conway 1971, p. 242). Several other maximal subgroups of Co₀ are found in this way. Moreover, two sporadic groups appear in the resulting chain.

There is a subgroup 2.A₈ × S₄, the only one of this chain not maximal in Co₀. Next there is the subgroup (2.A₇ × PSL₂(7)):2. Next comes (2.A₆ × SU₃(3)):2. The unitary group SU₃(3) (order 6,048) possesses a graph of 36 vertices, in anticipation of the next subgroup. That subgroup is (2.A₅ o 2.HJ):2, in which the Hall–Janko group HJ makes its appearance. The aforementioned graph expands to the Hall–Janko graph, with 100 vertices. Next comes (2.A₄ o 2.G₂(4)):2, G₂(4) being an exceptional group of Lie type.

The chain ends with 6.Suz:2 (Suz=Suzuki sporadic group), which, as mentioned above, respects a complex representation of the Leech Lattice.

Conway and Norton suggested in their 1979 paper that monstrous moonshine is not limited to the monster. Larissa Queen and others subsequently found that one can construct the expansions of many Hauptmoduln from simple combinations of dimensions of sporadic groups. For the Conway groups, the relevant McKay–Thompson series is ${\displaystyle T_{2A}(\tau )}$ = {1, 0, 276, −2,048, 11,202, −49,152, ...} (OEIS: A007246) and ${\displaystyle T_{4A}(\tau )}$ = {1, 0, 276, 2,048, 11,202, 49,152, ...} (OEIS: A097340) where one can set the constant term a(0) = 24,

${\displaystyle {\begin{aligned}j_{4A}(\tau )&=T_{4A}(\tau )+24\\&=\left({\frac {\eta ^{2}(2\tau )}{\eta (\tau )\,\eta (4\tau )}}\right)^{24}\\&=\left(\left({\frac {\eta (\tau )}{\eta (4\tau )}}\right)^{4}+4^{2}\left({\frac {\eta (4\tau )}{\eta (\tau )}}\right)^{4}\right)^{2}\\&={\frac {1}{q}}+24+276q+2048q^{2}+11202q^{3}+49152q^{4}+\dots \end{aligned}}}$

and η(τ) is the Dedekind eta function.

• Conway, John Horton (1968), "A perfect group of order 8,315,553,613,086,720,000 and the sporadic simple groups", Proceedings of the National Academy of Sciences of the United States of America, 61 (2): 398–400, Bibcode:1968PNAS...61..398C, doi:10.1073/pnas.61.2.398, MR 0237634, PMC 225171, PMID 16591697
• Brauer, R.; Sah, Chih-han, eds. (1969), Theory of finite groups: A symposium, W. A. Benjamin, Inc., New York-Amsterdam, MR 0240186
• Conway, John Horton (1969), "A group of order 8,315,553,613,086,720,000", The Bulletin of the London Mathematical Society, 1: 79–88, doi:10.1112/blms/1.1.79, ISSN 0024-6093, MR 0248216
• Conway, John Horton (1971), "Three lectures on exceptional groups", in Powell, M. B.; Higman, Graham (eds.), Finite simple groups, Proceedings of an Instructional Conference organized by the London Mathematical Society (a NATO Advanced Study Institute), Oxford, September 1969., Boston, MA: Academic Press, pp. 215–247, ISBN 978-0-12-563850-0, MR 0338152 Reprinted in Conway & Sloane (1999, 267–298)
• Conway, John Horton; Sloane, Neil J. A. (1999), Sphere Packings, Lattices and Groups, Grundlehren der Mathematischen Wissenschaften, vol. 290 (3rd ed.), Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4757-2016-7, ISBN 978-0-387-98585-5, MR 0920369
• Thompson, Thomas M. (1983), From error-correcting codes through sphere packings to simple groups, Carus Mathematical Monographs, vol. 21, Mathematical Association of America, ISBN 978-0-88385-023-7, MR 0749038
• Conway, John Horton; Parker, Richard A.; Norton, Simon P.; Curtis, R. T.; Wilson, Robert A. (1985), Atlas of finite groups, Oxford University Press, ISBN 978-0-19-853199-9, MR 0827219
• Griess, Robert L. Jr. (1998), Twelve sporadic groups, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-662-03516-0, ISBN 978-3-540-62778-4, MR 1707296
• Atlas of Finite Group Representations: Co₁ version 2
• Atlas of Finite Group Representations: Co₁ version 3
• Wilson, Robert A. (1983), "The maximal subgroups of Conway's group Co₁", Journal of Algebra, 85 (1): 144–165, doi:10.1016/0021-8693(83)90122-9, ISSN 0021-8693, MR 0723071
• Wilson, Robert A. (1988), "On the 3-local subgroups of Conway's group Co₁", Journal of Algebra, 113 (1): 261–262, doi:10.1016/0021-8693(88)90192-5, ISSN 0021-8693, MR 0928064
• Wilson, Robert A. (2009), The finite simple groups., Graduate Texts in Mathematics 251, vol. 251, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-84800-988-2, ISBN 978-1-84800-987-5, Zbl 1203.20012
• Witt, Ernst (1998), Collected papers. Gesammelte Abhandlungen, Springer Collected Works in Mathematics, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-642-41970-6, ISBN 978-3-540-57061-5, MR 1643949
• R. T. Curtis and B. T. Fairburn (2009), "Symmetric Representation of the elements of the Conway Group .0", Journal of Symbolic Computation, 44: 1044–1067.