Thompson sporadic group

Sporadic simple group

This article is about the sporadic simple group found by John G. Thompson. For the three unusual infinite groups F, T and V found by Richard Thompson, see Thompson groups.

Algebraic structure → Group theory
Group theory

Basic notions

* Subgroup
* Normal subgroup
* Group action

* Quotient group
* (Semi-)direct product
* Direct sum
* Free product
* Wreath product

Group homomorphisms

* kernel
* image

* simple
* finite
* infinite
* continuous
* multiplicative
* additive
* cyclic
* abelian
* dihedral
* nilpotent
* solvable

* Glossary of group theory
* List of group theory topics

Finite groups

* Cyclic group Zn
* Symmetric group Sn
* Alternating group An

* Dihedral group Dn
* Quaternion group Q

* Cauchy's theorem
* Lagrange's theorem

* Sylow theorems
* Hall's theorem

* p-group
* Elementary abelian group

* Frobenius group

* Schur multiplier

Classification of finite simple groups

* cyclic
* alternating
* Lie type
* sporadic

* Discrete groups
* Lattices

* Integers ( Z {\displaystyle \mathbb {Z} } )
* Free group

Modular groups

* PSL(2,  Z {\displaystyle \mathbb {Z} } )
* SL(2,  Z {\displaystyle \mathbb {Z} } )

* Arithmetic group
* Lattice
* Hyperbolic group

Topological and Lie groups

* Solenoid
* Circle

* General linear GL(n)

* Special linear SL(n)

* Orthogonal O(n)

* Euclidean E(n)

* Special orthogonal SO(n)

* Unitary U(n)

* Special unitary SU(n)

* Symplectic Sp(n)

* G2
* F4
* E6
* E7
* E8

* Lorentz
* Poincaré
* Conformal

* Diffeomorphism
* Loop

Infinite dimensional Lie group

* O(∞)
* SU(∞)
* Sp(∞)

Algebraic groups

* Linear algebraic group

* Reductive group

* Abelian variety

* Elliptic curve

* v
* t
* e

In the area of modern algebra known as group theory, the Thompson group Th is a sporadic simple group of order

90,745,943,887,872,000
= 215 * 310 * 53 * 72 * 13 * 19 * 31
≈ 9×1016.

## History

[edit]

Th is one of the 26 sporadic groups and was found by John G. Thompson (1976) and constructed by Smith (1976). They constructed it as the automorphism group of a certain lattice in the 248-dimensional Lie algebra of E8. It does not preserve the Lie bracket of this lattice, but does preserve the Lie bracket mod 3, so is a subgroup of the Chevalley group E8(3). The subgroup preserving the Lie bracket (over the integers) is a maximal subgroup of the Thompson group called the Dempwolff group (which unlike the Thompson group is a subgroup of the compact Lie group E8).

## Representations

[edit]

The centralizer of an element of order 3 of type 3C in the Monster group is a product of the Thompson group and a group of order 3, as a result of which the Thompson group acts on a vertex operator algebra over the field with 3 elements. This vertex operator algebra contains the E8 Lie algebra over F3, giving the embedding of Th into E8(3).

The full normalizer of a 3C element in the Monster group is S3 × Th, so Th centralizes 3 involutions alongside the 3-cycle. These involutions are centralized by the Baby monster group, which therefore contains Th as a subgroup.

The Schur multiplier and the outer automorphism group of the Thompson group are both trivial.

## Generalized monstrous moonshine

[edit]

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 Th, the relevant McKay-Thompson series is  T 3 C ( τ ) {\displaystyle T_{3C}(\tau )} (OEIS: A007245),

T 3 C ( τ ) = ( j ( 3 τ ) ) 1 / 3 = 1 q + 248 q 2 + 4124 q 5 + 34752 q 8 + 213126 q 11 + 1057504 q 14 + ⋯ {\displaystyle T_{3C}(\tau )={\Big (}j(3\tau ){\Big )}^{1/3}={\frac {1}{q}}\,+\,248q^{2}\,+\,4124q^{5}\,+\,34752q^{8}\,+\,213126q^{11}\,+\,1057504q^{14}+\cdots \,}

and j(τ) is the j-invariant.

## Maximal subgroups

[edit]

Linton (1989) found the 16 conjugacy classes of maximal subgroups of Th as follows:

Maximal subgroups of Th No. | Structure | Order | Index | Comments

|
|
|
|

1 | 3D4(2) : 3 | 634,023,936
= 212·35·72·13 | 143,127,000
= 23·35·53·19·31 |
2 | 25 ·L5(2) | 319,979,520
= 215·32·5·7·31 | 283,599,225
= 38·52·7·13·19 | the Dempwolff group
3 | 21+8
+ ·A9 | 92,897,280
= 215·34·5·7 | 976,841,775
= 36·52·7·13·19·31 | centralizer of involution
4 | U3(8) : 6 | 33,094,656
= 210·35·7·19 | 2,742,012,000
= 25·35·53·7·13·31 |
5 | (3 x G2(3)) : 2 | 25,474,176
= 27·37·7·13 | 3,562,272,000
= 28·33·53·7·19·31 | normalizer of a subgroup of order 3 (class 3A)
6 | (33 × 31+2
+) · 31+2
+ : 2S4 | 944,784
= 24·310 | 96,049,408,000
= 211·53·72·13·19·31 | normalizer of a subgroup of order 3 (class 3B)
7 | 32 · 37 : 2S4 | 944,784
= 24·310 | 96,049,408,000
= 211·53·72·13·19·31 |
8 | (3 × 34 : 2 · A6) : 2 | 349,920
= 25·37·5 | 259,333,401,600
= 210·33·52·72·13·19·31 | normalizer of a subgroup of order 3 (class 3C)
9 | 51+2
+ : 4S4 | 12,000
= 25·3·53 | 7,562,161,990,656
= 210·39·72·13·19·31 | normalizer of a subgroup of order 5
10 | 52 : GL2(5) | 12,000
= 25·3·53 | 7,562,161,990,656
= 210·39·72·13·19·31 |
11 | 72 : (3 × 2S4) | 7,056
= 24·32·72 | 12,860,819,712,000
= 211·38·53·13·19·31 |
12 | L2(19) : 2 | 6,840
= 23·32·5·19 | 13,266,950,860,800
= 212·38·52·72·13·31 |
13 | L3(3) | 5,616
= 24·33·13 | 16,158,465,792,000
= 211·37·53·72·19·31 |
14 | M10 | 720
= 24·32·5 | 126,036,033,177,600
= 211·38·52·72·13·19·31 |
15 | 31 : 15 | 465
= 3·5·31 | 195,152,567,500,800
= 215·39·52·72·13·19 |
16 | S5 | 120
= 23·3·5 | 756,216,199,065,600
= 212·39·52·72·13·19·31 |

## References

[edit]

* Linton, Stephen A. (1989), "The maximal subgroups of the Thompson group", Journal of the London Mathematical Society, Second Series, 39 (1): 79–88, doi:10.1112/jlms/s2-39.1.79, ISSN 0024-6107, MR 0989921
* Smith, P. E. (1976), "A simple subgroup of M? and E8(3)", The Bulletin of the London Mathematical Society, 8 (2): 161–165, doi:10.1112/blms/8.2.161, ISSN 0024-6093, MR 0409630
* Thompson, John G. (1976), "A conjugacy theorem for E8", Journal of Algebra, 38 (2): 525–530, doi:10.1016/0021-8693(76)90235-0, ISSN 0021-8693, MR 0399193

## External links

[edit]

* MathWorld: Thompson group
* Atlas of Finite Group Representations: Thompson group

*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template