Thompson sporadic group

For the unrelated infinite simple groups found by Richard Thompson, see Thompson groups.

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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(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:

• $2_+^{1+8}\cdot A_9$,
• $2^5\cdot L_5(2)$,
• $(3\times G_2(3)): 2$,
• $(3^3\times 3_+^{1+2})\cdot 3_+^{1+2}: 2S_4$,
• $3^2\cdot 3^7:2S_4$,
• $(3\times 3^4:2\cdot A_6): 2$,
• $5_+^{1+2}: 4S_4$,
• 5²:GL(2,5),
• $7^2:(3\times 2S_4)$,
• 31:15,
• ³D₄(2):3,
• U₃(8):6,
• L₂(19),
• L₃(3),
• M₁₀,
• S₅.

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