Conway group

From Academic Kids

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

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[ei/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).

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.

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.
  • 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
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