Higman–Sims group

HS is one of the 26 sporadic groups and was found by Donald G. Higman and Charles C. Sims (1968). They were attending a presentation by Marshall Hall on the Hall–Janko group J₂. It happens that J₂ is a permutation group of 100 points, and the stabilizer of a point is a subgroup with two other orbits of lengths 36 and 63. Inspired by this they decided to check for other rank 3 permutation groups on 100 points. They soon focused on a possible one containing the Mathieu group M₂₂, which has permutation representations on 22 and 77 points. (The latter representation arises because the M₂₂ Steiner system has 77 blocks.) By putting together these two representations, they found HS, with a one-point stabilizer isomorphic to M₂₂.

HS is 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 HS is a transitive group of permutations of a 100 element set.

Graham Higman (1969) independently discovered the group as a doubly transitive permutation group acting on a certain 'geometry' on 176 points.

The Schur multiplier has order 2, the outer automorphism group has order 2, and the group 2.HS.2 appears as an involution centralizer in the Harada–Norton group.

Generalized Monstrous Moonshine

Conway and Norton suggested in their 1979 paper that monstrous moonshine is not limited to the monster, but that similar phenomena may be found for other groups. Larissa Queen and others subsequently found that one can construct the expansions of many Hauptmoduln from simple combinations of dimensions of sporadic groups. For HS, the McKay-Thompson series is $T_{10A}(\tau)$ where one can set a(0) = 4 ( A058097),

\begin{align}j_{10A}(\tau) &=T_{10A}(\tau)+4\\ &=\Big(\big(\tfrac{\eta(\tau)\,\eta(5\tau)}{\eta(2\tau)\,\eta(10\tau)}\big)^{2}+2^2 \big(\tfrac{\eta(2\tau)\,\eta(10\tau)}{\eta(\tau)\,\eta(5\tau)}\big)^{2}\Big)^2\\ &=\Big(\big(\tfrac{\eta(\tau)\,\eta(2\tau)}{\eta(5\tau)\,\eta(10\tau)}\big)+5 \big(\tfrac{\eta(5\tau)\,\eta(10\tau)}{\eta(\tau)\,\eta(2\tau)}\big)\Big)^2-4\\ &=\frac{1}{q} + 4 + 22q + 56q^2 +177q^3+352q^4+870q^5+1584q^6+\dots \end{align}

Relationship with the Conway groups

Conway (1968) showed how the Higman–Sims graph could be embedded in the Leech lattice. Here, HS fixes a 2-3-3 triangle and a 22-dimensional sublattice. The Higman–Sims group thus becomes a subgroup of each of the Conway groups Co₀, Co₂ and Co₃. If a conjugate of HS in Co₀ fixes a particular point of type 3, this point is found in 276 triangles of type 2-2-3, which this copy of HS permutes in orbits of 176 and 100. This also provides a 22 dimensional representation of HS, acting on a 22 dimensional lattice given by the orthogonal complement of a 2-3-3 triangle with one vertex at the origin.

Maximal subgroups

Magliveras (1971) showed that HS has 12 conjugacy classes of maximal subgroups.

M₂₂, order 443520
U₃(5):2, order 252000 – one-point stabilizer in doubly transitive representation of degree 176
U₃(5):2 – conjugate to class above in HS:2
PSL(3,4):2, order 40320
S₈, order 40320
2⁴.S₆, order 11520
4³:PSL(3,2), order 10752
M₁₁, order 7920
M₁₁ – conjugate to class above in HS:2
4.2⁴.S₅, order 7680 – centralizer of involution moving 80 vertices of Higman–Sims graph
2 × A₆.2², order 2880 – centralizer of involution moving all 100 vertices
5:4 × A₅, order 1200

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, ISSN 0027-8424, MR 0237634
Dixon, John D.; Mortimer, Brian (1996), Permutation groups, Graduate Texts in Mathematics 163, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94599-6, MR 1409812
Gallian, Joseph (1976), "The search for finite simple groups", Mathematics Magazine 49 (4): 163–180, doi:10.2307/2690115, ISSN 0025-570X, JSTOR 2690115, MR 0414688
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
Higman, Donald G.; Sims, Charles C. (1968), "A simple group of order 44,352,000", Mathematische Zeitschrift 105 (2): 110–113, doi:10.1007/BF01110435, ISSN 0025-5874, MR 0227269
Higman, Graham (1969), "On the simple group of D. G. Higman and C. C. Sims", Illinois Journal of Mathematics 13: 74–80, ISSN 0019-2082, MR 0240193
Magliveras, Spyros S. (1971), "The subgroup structure of the Higman–Sims simple group", Bulletin of the American Mathematical Society 77 (4): 535–539, doi:10.1090/S0002-9904-1971-12743-X, ISSN 0002-9904, MR 0283077

Higman–Sims group

History

Generalized Monstrous Moonshine

Relationship with the Conway groups

Maximal subgroups

References

External links