Conway group

In mathematics, the Conway groups Co₁, Co₂ and Co₃ are the three sporadic groups discovered by John Horton Conway.

The largest of the Conway groups, Co₁, is of order

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

and is obtained as the quotient of Co₀ (group of automorphisms of the Leech lattice Λ that fix the origin) 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.

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[e^2πi/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.

References

• 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, doi:10.1073/pnas.61.2.398, MR 0237634 
• 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, ISSN 0024-6093, MR 0248216 
• Conway, John Horton (1971), "Three lectures on exceptional groups", in Powell, M. B.; Higman, Graham, 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 290 (3rd ed.), Berlin, New York: Springer-Verlag, 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 21, Mathematical Association of America, ISBN 978-0-88385-023-7, MR 749038 
• 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 827219 
• Griess, Robert L. Jr. (1998), Twelve sporadic groups, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, 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 723071 
• 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 928064 
• Wilson, Robert A. (2009), The finite simple groups., Graduate Texts in Mathematics 251, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-84800-988-2, ISBN 978-1-84800-987-5, Zbl 05622792