Thompson sporadic group
| Algebraic structure → Group theory Group theory |
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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(F3). 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(F3).)
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.
References
- 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, MR0409630
- 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, MR0399193