Conway group

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

All are closely related to the Leech lattice Λ. The largest, Co₁, of order

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

is obtained as the quotient of the automorphism group of Λ 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 1/16 of its norm, vˑv.) As the scalar −1 fixes no non-zero vector, we can regard these two groups as subgroups of Co₁.

[edit] Other sporadic groups

The groups Co₂ and Co₃ both contain the McLaughlin group McL (of order 898,128,000) and the Higman-Sims group (of order 44,352,000), which can be described as the pointwise stabilizers of a type

2-2-3 triangle

and a type

2-3-3 triangle,

respectively. 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 6-element group of complex scalar matrices, gives the Suzuki group Suz (of order 448,345,497,600). Suz is the only proper sporadic subgroup of Co₁ that retains 13 as a prime factor.

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 7 simple groups described above comprise what Robert Griess calls the second generation of the Happy Family, the latter being the sporadic simple groups found within the Monster group. Several of the 7 groups contain at least some of the 5 Mathieu groups, which comprise the first generation.

[edit] References

• Conway, J. H.: A perfect group of order 8,315,553,613,086,720,000 and the sporadic simple groups, Proc. Nat. Acad. Sci. U.S.A. 61 (1968), 398-400.
• Thompson, Thomas M.: "From Error Correcting Codes through Sphere Packings to Simple Groups", Carus Mathematical Monographs, Mathematical Association of America, 1983.
• Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A., Atlas of finite groups. Maximal subgroups and ordinary characters for simple groups. With computational assistance from J. G. Thackray. Eynsham: Oxford University Press, 1985, ISBN 0-19-853199-0
• Griess, Robert L.: "Twelve Sporadic Groups", Springer-Verlag, 1998.