Higman–Sims group

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.

History

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 acts as a permutation group on the Hall-Janko graph of 100 points, the stabilizer of one point being 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.

Relationship to Conway groups

Conway (1968) identified the Higman–Sims group as a subgroup of the Conway group Co₀. In Co₀ HS arises as a pointwise stabilizer of a 2-3-3 triangle, one whose vertices are type 2 and 3 points whose sum is the origin. HS thus is a subgroup of each of the Conway groups Co₀, Co₂ and Co₃.

In the Leech lattice, suppose a type 3 point v is fixed by an instance of Co₃. Count the type 3 points w such that u+w is type 2. Wilson (2009) (p. 208) shows their number is 11,178=2*3⁵*23. He also shows this Co₃ is transitive on these w. Hence the subgroup HS is well-defined.

|HS| = |Co₃|/11,178 = 44,352,000.

In fact |HS| = 100|M₂₂| and there are instances of HS including a permutation matrix representation of the Mathieu group M₂₂.

If an instance 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 fact leads to Graham Higman's construction as well as to the Higman-Sims graph.

A 2-3-3 triangle defines a 2-dimensional subspace fixed pointwise by HS. The standard representation of HS can thus be reduced to a 22-dimensional one

A Higman-Sims graph

Wilson (2009) (p. 210) gives an example of a Higman-Sims graph within the Leech lattice, permuted by the representation of M₂₂ on the last 22 coordinates:

22 points of shape (1,1,-3,1²¹)
77 points of shape (2,2,2⁶,0¹⁶)
A 100th point (4,4,0²²)

Differences of adjacent points are of type 3; those of non-adjacent ones are of type 2. Here, HS fixes a 2-3-3 triangle with vertices (-4,4,0²²), (5,1²³), and -(1,5,1²²).

Maximal subgroups

Magliveras (1971) found the 12 conjugacy classes of maximal subgroups of HS as follows:

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

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}

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
Wilson, Robert A. (2009), The finite simple groups., Graduate Texts in Mathematics 251, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-84800-988-2, ISBN 978-1-84800-987-5, Zbl 05622792

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