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 = 2⁹3²5³711

: ≈ 4.

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 . 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.

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.

The Higman-Sims group can be constructed with the following two generators: 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

found the 12 conjugacy classes of maximal subgroups of HS as follows:

{| class="wikitable"

|+ Maximal subgroups of HS

|-

! No.||Subgroup||Order||Index||Orbits on Higman-Sims graph||Comments

|-

| 1 ||M₂₂||align=right|443,520<br />=&nbsp;2⁷·3²·5·7·11 ||align=right| 100<br />=&nbsp;2²·5² || 1, 22, 77 || one-point stabilizer on Higman-Sims graph

|-

| 2,3||U₃(5):2 ||align=right|252,000<br />=&nbsp;2⁵·3²·5³·7||align=right| 176<br />=&nbsp;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 ||align=right| 40,320<br />=&nbsp;2⁷·3²·5·7 ||align=right| 1,100<br />=&nbsp;2²·5²·11 || 2, 42, 56 || stabilizer of edge

|-

| 5 ||S₈ ||align=right| 40,320<br />=&nbsp;2⁷·3²·5·7 ||align=right| 1,100<br />=&nbsp;2²·5²·11 || 30, 70 || centralizer of an outer automorphism of order 2 (class 2C)

|-

| 6 ||2⁴.S₆ ||align=right| 11,520<br />=&nbsp;2⁸·3²·5 ||align=right| 3,850<br />=&nbsp;2·5²·7·11 || 2, 6, 32, 60 || stabilizer of non-edge

|-

| 7 ||4³:L₃(2) ||align=right| 10,752<br />=&nbsp;2⁹·3·7 ||align=right| 4,125<br />=&nbsp;3·5³·11 || 8, 28, 64

|-

| 8,9||M₁₁||align=right| 7,920<br />=&nbsp;2⁴·3²·5·11 ||align=right| 5,600<br />=&nbsp;2⁵·5²·7 || 12, 22, 66 ||two classes, fused in HS:2

|-

| 10 ||4.2⁴.S₅ ||align=right| 7,680<br />=&nbsp;2⁹·3·5 ||align=right| 5,775<br />=&nbsp;3·5²·7·11 || 20, 80|| centralizer of involution (class 2A) moving 80 vertices of Higman–Sims graph

|-

| 11 ||2 × A₆.2² ||align=right| 2,880<br />=&nbsp;2⁶·3²·5 ||align=right|15,400<br />=&nbsp;2³·5²·7·11|| 40, 60|| centralizer of involution (class 2B) moving all 100 vertices

|-

| 12 ||5:4 × A₅ ||align=right| 1,200<br />=&nbsp;2⁴·3·5² ||align=right|36,960<br />=&nbsp;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. Listed are 2 permutation representations: on the 100 vertices of the Higman–Sims graph, and on the 176 points of Graham Higman's geometry.

{| class="wikitable" style="margin: 1em auto;"

|-

! 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||rowspan="2"|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||rowspan="2"|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 <math>T_{10A}(\tau)</math> where one can set (),

:<math>\begin{align} 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{align}</math>

References

• J. S. Frame (1972) 'Computations of Characters of the Higman-Sims Group and its Automorphism Group' Journal of Algebra, 20, 320-349

External links

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