| Algebraic structure → Group theory Group theory |
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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.
| Algebraic structure → Group theory Group theory |
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In the area of modern algebra known as group theory, the Thompson groupTh is a sporadic simple group of order
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).
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 nomralizer 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.
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 ( OEIS: A007245 ),
and j(τ) is the j-invariant.
Linton (1989) found the 16 conjugacy classes of maximal subgroups of Th as follows:
In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group, having order
808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000
= 246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71
≈ 8×1053.
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 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 initial numerical observation was made by John McKay in 1978, and the phrase 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 mathematics, a Ree group is a group of Lie type over a finite field constructed by Ree from an exceptional automorphism of a Dynkin diagram that reverses the direction of the multiple bonds, generalizing the Suzuki groups found by Suzuki using a different method. They were the last of the infinite families of finite simple groups to be discovered.
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 groupJ2 or the Hall-Janko groupHJ 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′ or F3+ 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
