Last updated

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

## Contents

215 ·310 ·53 ·72 ·13 ·19 ·31
= 90745943887872000
≈ 9×1016.

## History

Th is one of the 26 sporadic groups and was found by John G.Thompson  ( 1976 ) and constructed by Geoff Smith. 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

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

## 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 Th, the relevant McKay-Thompson series is $T_{3C}(\tau )$ (),

$T_{3C}(\tau )={\Big (}j(3\tau ){\Big )}^{1/3}={\frac {1}{q}}\,+\,248q^{2}\,+\,4124q^{5}\,+\,34752q^{8}\,+\,213126q^{11}\,+\,1057504q^{14}+\cdots \,$ and j(τ) is the j-invariant.

## Maximal subgroups

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

• 2+1+8 · A
• 25 · L5(2)   This is the Dempwolff group
• (3 x G2(3)) : 2
• (33 × 3+1+2) · 3+1+2 : 2S4
• 32 · 37 : 2S4
• (3 × 34 : 2 · A6) : 2
• 5+1+2 : 4S4
• 52 : GL2(5)
• 72 : (3 × 2S4)
• 31 : 15
• 3D4(2) : 3
• U3(8) : 6
• L2(19)
• L3(3)
• M10
• S5

## Related Research Articles In the area of modern algebra known as group theory, the baby monster groupB (or, more simply, the baby monster) is a sporadic simple group of order In the area of modern algebra known as group theory, the Conway groups are the three sporadic simple groups Co1, Co2 and Co3 along with the related finite group Co0 introduced by (Conway 1968, 1969). In the area of modern algebra known as group theory, the Suzuki groupSuz or Sz is a sporadic simple group of order In group theory, the Tits group2F4(2)′, named for Jacques Tits (French: [tits]), is a finite simple group of order In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group M and modular functions, in particular, the j function. The term was coined by John Conway and Simon P. Norton in 1979. In the area of modern algebra known as group theory, the Held groupHe is a sporadic simple group of order In the area of abstract algebra known as group theory, the O'Nan groupO'N or O'Nan–Sims group is a sporadic simple group of order In the area of modern algebra known as group theory, the Rudvalis groupRu is a sporadic simple group of order In the area of modern algebra known as group theory, the Harada–Norton groupHN is a sporadic simple group of order In the area of modern algebra known as group theory, the Janko groupJ1 is a sporadic simple group of order In the area of modern algebra known as group theory, the Janko groupJ4 is a sporadic simple group of order In the area of modern algebra known as group theory, the McLaughlin group McL is a sporadic simple group of order

In mathematical finite group theory, the Dempwolff group is a finite group of order 319979520 = 215·32·5·7·31, that is the unique nonsplit extension of by its natural module of order . The uniqueness of such a nonsplit extension was shown by Dempwolff (1972), and the existence by Thompson (1976), who showed using some computer calculations of Smith (1976) that the Dempwolff group is contained in the compact Lie group as the subgroup fixing a certain lattice in the Lie algebra of , and is also contained in the Thompson sporadic group (the full automorphism group of this lattice) as a maximal subgroup. In the area of modern algebra known as group theory, the Fischer groupFi24 or F24′ is a sporadic simple group of order In the area of modern algebra known as group theory, the Fischer groupFi23 is a sporadic simple group of order In the area of modern algebra known as group theory, the Fischer groupFi22 is a sporadic simple group of order In the area of modern algebra known as group theory, the Conway groupCo2 is a sporadic simple group of order In the area of modern algebra known as group theory, the Conway group is a sporadic simple group of order In the area of modern algebra known as group theory, the Conway groupCo1 is a sporadic simple group of order

• 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:, ISSN   0021-8693, MR   0399193