Thompson sporadic group

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

• 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, MR989921
• 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

External links

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