Thompson sporadic group

In the area of modern algebra known as group theory, the Thompson group Th is a sporadic simple group of order

   2¹⁵ · 3¹⁰ · 5³ · 7² · 13 · 19 · 31

= 90745943887872000

≈ 9×10¹⁶.

History

Th is one of the 26 sporadic groups and was found by John G. Thompson (1976) and constructed by Geoff Smith. 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).
Representations

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.
Generalized Monstrous Moonshine

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 \( T_{{3C}}(\tau ) \) (OEIS: A007245),

\( T_{{3C}}(\tau )={\Big (}j(3\tau ){\Big )}^{{1/3}}={\frac {1}{q}}\,+\,248q^{2}\,+\,4124q^{5}\,+\,34752q^{8}\,+\,213126q^{{11}}\,+\,1057504q^{{14}}+\cdots \, \)

and j(τ) is the j-invariant.
Maximal subgroups

Linton (1989) found the 16 conjugacy classes of maximal subgroups of Th as follows:

• 2₊¹⁺⁸ · A
• 2⁵ · L₅(2)   This is the Dempwolff group
• (3 x G₂(3)) : 2
• (3³ × 3₊¹⁺²) · 3₊¹⁺² : 2S₄
• 3² · 3⁷ : 2S₄
• (3 × 3⁴ : 2 · A₆) : 2
• 5₊¹⁺² : 4S₄
• 5² : GL₂(5)
• 7² : (3 × 2S₄)
• 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, MR 0989921
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, MR 0409630
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, MR 0399193

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