Conway group

In mathematics, the Conway groups Co0, Co1, Co2 and Co3 are four groups discovered by John Horton Conway. The last three are sporadic groups: finite simple groups that are not part of infinite series.

The largest of the Conway groups, Co0, is the group of automorphisms of the Leech lattice Λ. It has order

8,315,553,613,086,720,000

but it is not a simple group. The group Co1 has order

4,157,776,806,543,360,000

and it is obtained as the quotient of Co0 by its center, which consists of the scalar matrices ±1. The groups Co2 (of order 42,305,421,312,000) and Co3 (of order 495,766,656,000) consist of the automorphisms of Λ fixing a lattice vector of type 2 and a vector of type 3 respectively. (The type of a vector is half of its square norm, v·v.) As the scalar −1 fixes no non-zero vector, these two groups are isomorphic to subgroups of Co1.

History

Thomas Thompson (1983) relates how John Leech about 1964 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 (the double cover of Conway's largest simple group).

Other sporadic groups

Conway and Thompson found that four recently discovered sporadic simple groups, described in the conference proceedings (Brauer & Sah 1969), were isomorphic to subgroups or quotients of subgroups of Co1.

Two of these (subgroups of Co2 and Co3) can be defined as pointwise stabilizers of triangles with vertices, of sum zero, of types 2 and 3. A 2-2-3 triangle is fixed by the McLaughlin group McL (order 898,128,000). A 2-3-3 triangle is fixed by the Higman-Sims group (order 44,352,000).

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

Z[e2πi/3]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 (of order 448,345,497,600). This group was discovered by Michio Suzuki in 1968.

A similar construction gives the Hall-Janko group J2 (of 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.

Generalized Monstrous Moonshine

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 T_{2A}(\tau) = {1, 0, 276, -2048, 11202, -49152,..}( A007246) and T_{4A}(\tau) = {1, 0, 276, 2048, 11202, 49152,..}( A097340) where one can set the constant term a(0) = 24,

\begin{align}j_{4A}(\tau) &=T_{4A}(\tau)+24\\ &=\Big(\tfrac{\eta^2(2\tau)}{\eta(\tau)\,\eta(4\tau)} \Big)^{24} \\ &=\Big(\big(\tfrac{\eta(\tau)}{\eta(4\tau)}\big)^{4}+4^2 \big(\tfrac{\eta(4\tau)}{\eta(\tau)}\big)^{4}\Big)^2\\ &=\frac{1}{q} + 24+ 276q + 2048q^2 +11202q^3+49152q^4+\dots \end{align}

and η(τ) is the Dedekind eta function.

References