In the mathematical field of group theory, the Thompson group Th, found by John G. Thompson (1976) and constructed by Smith (1976), is a sporadic simple group of order
- 215 · 310 · 53 · 72 · 13 · 19 · 31
- = 90745943887872000
- ≈ 9 · 1016.
Thompson and Smith constructed the Thompson group as the group of automorphisms 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 Schur multiplier and the outer automorphism group of the Thompson group are both trivial.
The Thompson group contains the Dempwolff group as a maximal subgroup.
Linton (1989) found the 16 classes of maximal subgroups of the Thompson group, as follows:
- ,
- ,
- ,
- ,
- ,
- ,
- ,
- ,
- ,
- ,
- ,
- ,
- ,
- ,
- ,
- .
Famous quotes containing the words thompson and/or group:
“She went her unremembering way,
She went and left in me
The pang of all the partings gone,
And partings yet to be.”
—Francis Thompson (1859–1907)
“He hung out of the window a long while looking up and down the street. The world’s second metropolis. In the brick houses and the dingy lamplight and the voices of a group of boys kidding and quarreling on the steps of a house opposite, in the regular firm tread of a policeman, he felt a marching like soldiers, like a sidewheeler going up the Hudson under the Palisades, like an election parade, through long streets towards something tall white full of colonnades and stately. Metropolis.”
—John Dos Passos (1896–1970)