Conway group

Conway group

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

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.

All are closely related to the Leech lattice Λ. The largest, Co₁ (of order 8,315,553,613,086,720,000), is obtained by dividing 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 length 2 and a vector of √6 respectively. As the scalar -1 fixes no non-zero vector, we can regard these two groups as subgroups of Co₁.

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 2-2-√6 triangle and a 2-√6-√6 triangle respectively. Identifying R²⁴ with C¹² and Λ with Z[e^2πi/3]¹², the resulting auotmorphism 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). 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.

See:

Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A. (1985). Atlas of finite groups. Maximal subgroups and ordinary characters for simple groups. With computational assistance from J. G. Thackray. Eynsham: Oxford University Press. ISBN 0-19-853199-0