Higman–Sims group

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

   44,352,000 = 2⁹ · 3² · 5³ · 7 · 11

≈ 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.

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. 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.

Construction

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)

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 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.

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 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.

Two classes of involutions

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.

Maximal subgroups

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

Maximal subgroups of HS

No.	Subgroup	Order	Index	Orbits on Higman-Sims graph	Comments
1	M22	443,520 = 27·32·5·7·11	100 = 22·52	1, 22, 77	one-point stabilizer on Higman-Sims graph
2,3	U3(5):2	252,000 = 25·32·53·7	176 = 24·11	imprimitive on pair of Hoffman-Singleton graphs of 50 vertices each	one-point stabilizer in doubly transitive representation of degree 176; two classes, fused in HS:2
4	L3(4).2	40,320 = 27·32·5·7	1,100 = 22·52·11	2, 42, 56	stabilizer of edge
5	S8	40,320 = 27·32·5·7	1,100 = 22·52·11	30, 70	centralizer of an outer automorphism of order 2 (class 2C)
6	24.S6	11,520 = 28·32·5	3,850 = 2·52·7·11	2, 6, 32, 60	stabilizer of non-edge
7	43:L3(2)	10,752 = 29·3·7	4,125 = 3·53·11	8, 28, 64
8,9	M11	7,920 = 24·32·5·11	5,600 = 25·52·7	12, 22, 66	two classes, fused in HS:2
10	4.24.S5	7,680 = 29·3·5	5,775 = 3·52·7·11	20, 80	centralizer of involution (class 2A) moving 80 vertices of Higman–Sims graph
11	2 × A6.22	2,880 = 26·32·5	15,400 = 23·52·7·11	40, 60	centralizer of involution (class 2B) moving all 100 vertices
12	5:4 × A5	1,200 = 24·3·52	36,960 = 25·3·5·7·11	imprimitive on 5 blocks of 20	normalizer of a subgroup of order 5 (class 5B)

Conjugacy classes

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]

Class	Order of centralizer	No. elements	Trace	On 100	On 176
1A	44,352,000	1 = 1	24
2A	7,680	5775 = 3 · 52 · 7 · 11	8	120,240	116,280
2B	2,880	15400 = 23 · 52 · 5 · 7 · 11	0	250	112, 282
3A	360	123200 = 26 · 52 · 7 · 11	6	110,330	15,357
4A	3,840	11550 = 2 · 3 · 52 · 7 · 11	-4	210420	116,440
4B	256	173250 = 2 · 32 · 53 · 7 · 11	4	18,26,420	28,440
4C	64	693000 = 23 · 32 · 53 · 7 · 11	4	14,28,420	14,26,440
5A	500	88704 = 27 · 32 · 7 · 11	-1	520	1,535
5B	300	147840 = 27 · 3 · 5 · 7 · 11	4	520	16,534
5C	25	1774080 = 29 · 32 · 5 · 7	4	15,519	1,535
6A	36	1232000 = 27 · 53 · 7 · 11	0	25,615	13,2,33,627
6B	24	1848000 = 26 · 3 · 53 · 7 · 11	2	12,24,36,612	1, 22,35,626
7A	7	6336000 = 29 · 32 · 53 · 11	3	12,714	1,725
8A	16	2772000 = 25 · 32 · 53 · 7 · 11	2	12,23,43,810	44, 820
8B	16	2772000 = 25 · 32 · 53 · 7 · 11	2	22,44,810	12,2,43,820
8C	16	2772000 = 25 · 32 · 53 · 7 · 11	2	22,44,810	12 2, 43, 820
10A	20	2217600 = 27 · 32 · 52 · 7 · 11	3	54,108	1,53,1016
10B	20	2217600 = 27 · 32 · 52 · 7 · 11	0	1010	12,22,52,1016
11A	11	4032000 = 29 · 32 · 53 · 7	2	11119	1116	Power equivalent
11B	11	4032000 = 29 · 32 · 53 · 7	2	11119	1116
12A	12	3696000 = 27 · 3 · 53 · 7 · 11	2	21,42,63,126	1,35,4,1213
15A	15	2956800 = 29 · 3 · 52 · 7 · 11	1	52,156	32,5,1511
20A	20	2217600 = 27 · 32 · 52 · 7 · 11	1	102,204	1,53,208	Power equivalent
20B	20	2217600 = 27 · 32 · 52 · 7 · 11	1	102,204	1,53,208

Generalized Monstrous Moonshine

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),

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