Higman–Sims group

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In the area of modern algebra known as group theory, the Higman–Sims group HS is a sporadic simple group of order

   2⁹⋅3²⋅5³⋅7⋅11 = 44352000

≈ 4×10⁷.

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.

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. The smallest faithful complex representation of HS has dimension 22.^[1]

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

GAP code to build the Higman-Sims group is presented as an example in the GAP documentation itself.^[2]

The Higman-Sims group can be constructed with the following two generators:^[2]

(1,50,65) (2,89,62,52,88,25) (3,46,57,18,74,55) (4,45,10,70,56,39) (5,97,77) (6,84,8,48,99,67) (7,26,92,28,20,100) (9,30,79,66,49,95) (11,72) (12,94,98,27,83,93) (13,31,61,59,40,47) (14,51,68,44,16,34) (15,38) (17,82,87) (19,76,73,71,63,32) (21,37,58,69,75,35) (22,53,81) (23,33,54) (24,43,80,78,29,86) (42,64) (60,90,96) (85,91)

and

(1,65,44,13,34,57) (2,10,39,54,42,84) (3,15,69,63,37,11) (5,21,79) (6,89,49,64,46,80) (7,70,93,29,8,38) (9,81,17,23,77,59) (12,68,66,75,96,82) (14,18,95,43,76,32) (16,33,99,26,92,48) (19,50) (20,97,83) (22,88,85,53,24,56) (25,62,67) (27,98) (28,55) (30,58,71,86,94,90) (31,87,52,78,100,60) (35,61,51) (36,73,72) (40,74) (41,45,47)

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 edges (differences of vertices) are type 2 and 3 vectors. HS thus is a subgroup of each of the Conway groups Co₀, Co₂ and Co₃.

Wilson (2009) (p. 208) shows that the group HS is well-defined. In the Leech lattice, suppose a type 3 point v is fixed by an instance of Co₃. Count the type 2 points w such that the inner product v·w = 2 (and thus v-w is type 3). He shows that their number is 11,178 = 2⋅3⁵⋅23 and that this Co₃ is transitive on these w.

|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 that 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. HS is doubly transitive on the 176 and rank 3 on the 100.

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.

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 x = (5, 1²³), y = (1, 5, 1²²), and z the origin. x and y are of type 3 while x-y = (4, −4, 0²²) is of type 2. Any vertex of the graph differs from x, y, and z by vectors of type 2.

An involution in the subgroup M₂₂ transposes 8 pairs of co-ordinates. As a permutation matrix in Co₀ it has trace 8. It can shown that it moves 80 of the 100 vertices of the Higman-Sims graph. No transposed pair of vertices is an edge in the graph.

There is another class of involutions, of trace 0, that move all 100 vertices.^[3] As permutations in the alternating group A₁₀₀, being products of an odd number (25) of double transpositions, these involutions lift to elements of order 4 in the double cover 2.A₁₀₀. HS thus has a double cover 2.HS, which is not related to the double cover of the subgroup M₂₂.

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

Traces of matrices in a standard 24-dimensional representation of HS are shown. ^[4] Listed are 2 permutation representations: on the 100 vertices of the Higman–Sims graph, and on the 176 points of Graham Higman's geometry.^[5]

Conway and Norton suggested in their 1979 paper that monstrous moonshine is not limited to the monster group, 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 ${\displaystyle T_{10A}(\tau )}$ where one can set a(0) = 4 (OEIS: A058097),

${\displaystyle {\begin{aligned}j_{10A}(\tau )&=T_{10A}(\tau )+4\\&=\left(\left({\frac {\eta (\tau )\;\eta (5\tau )}{\eta (2\tau )\;\eta (10\tau )}}\right)^{2}+2^{2}\left({\frac {\eta (2\tau )\;\eta (10\tau )}{\eta (\tau )\;\eta (5\tau )}}\right)^{2}\right)^{2}\\&=\left(\left({\frac {\eta (\tau )\;\eta (2\tau )}{\eta (5\tau )\;\eta (10\tau )}}\right)+5\left({\frac {\eta (5\tau )\;\eta (10\tau )}{\eta (\tau )\;\eta (2\tau )}}\right)\right)^{2}-4\\&={\frac {1}{q}}+4+22q+56q^{2}+177q^{3}+352q^{4}+870q^{5}+1584q^{6}+\cdots \end{aligned}}}$

• MathWorld: Higman–Sims
• Atlas of Finite Group Representations: Higman–Sims group