Higman–Sims group
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In mathematics, the Higman–Sims group is a finite sporadic simple group of order 44,352,000. It can be characterized as the simple subgroup of index two in the group of automorphisms of the Higman–Sims graph. The Higman–Sims graph has 100 nodes, so the Higman–Sims group, or HS, has a permutation representation of degree 100.
HS is named for Donald G. Higman and Charles C. Sims.
It is said that one day in 1967 Higman and Sims were attending a presentation by Marshall Hall on the Hall-Janko group, which has a representation of degree 100, with a subgroup with orbits of 36 and 63. It occurred to them to try the Mathieu group M22, which has representations of degree 22 and 77. The M22 Steiner system has 77 blocks. Quickly they found HS, with a one-point stabilizer isomorphic to M22.
"Higman" may also refer to the mathematician Graham Higman of the University of Oxford who simultaneously discovered the group as the automorphism group of a certain 'geometry' on 176 points. Consequently, HS has a doubly transitive representation of degree 176.
[edit] Relationship with the Conway Groups
In his now classic 1968 paper John Horton Conway showed how the Higman-Sims graph could be embedded in the Leech lattice. Here, HS fixes a 233 triangle and a 22 dimensional sublattice. The group thus becomes a subgroup of each of the Conway groups Co1, Co2 and Co3. If a conjugate of HS in Co1 fixes a particular point of type 3, this point is found in 276 triangles of type 223, which this copy of HS permutes in orbits of 176 and 100. This provides an explicit way of approaching a low dimensional representation of the group, and with it a straightforward means of computing inside the group.
[edit] References
- John H. Conway "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 USA S. 61 (2): 398. (1968)
- John D. Dixon & Brian Mortimer, 'Permutation Groups', Springer-Verlag (1996).
- Joseph A. Gallian, 'The Search for Finite Simple Groups', Mathematics Magazine, v. 48 (1976), no. 4, p. 163.
- Robert L. Griess, Jr, "Twelve Sporadic Groups", Springer-Verlag, 1998.
- Higman D.G. and Sims C.C. "A simple group of order 44,352,000" Zentralblatt-MATH 1O5 (1968), 110-113.
