Thompson group (finite)

In the area of modern algebra known as group theory, the Thompson group ''Th'' is a sporadic simple group of order
:   2¹⁵3¹⁰5³7²131931
: = 90745943887872000
: ≈ 9.

History

''Th'' is one of the 26 sporadic groups and was found by and constructed by . 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 $T_(\tau)$ (),

:$T_(\tau) = \Big(j(3\tau)\Big)^ = \frac\,+\,248q^2\,+\,4124q^5\,+\,34752q^8\,+\,213126q^\,+\,1057504q^+\cdots\,$

and ''j''(''τ'') is the j-invariant.

Maximal subgroups

found the 16 conjugacy classes of maximal subgroups of ''Th'' as follows:

* 2₊¹⁺⁸ · ''A''
* 2⁵ · ''L''₅(2) This is the Dempwolff group
* (3 x ''G''₂(3)) : 2
* (3³ × 3₊¹⁺²) · 3₊¹⁺² : 2''S''₄
* 3² · 3⁷ : 2''S''₄
* (3 × 3⁴ : 2 · ''A''₆) : 2
* 5₊¹⁺² : 4''S''₄
* 5² : ''GL''₂(5)
* 7² : (3 × 2''S''₄)
* 31 : 15
*³''D''₄(2) : 3
* ''U''₃(8) : 6
* ''L''₂(19)
* ''L''₃(3)
* ''M''₁₀
* ''S''₅

References

*
*
*{{Citation , last1=Thompson , first1=John G. , author1-link=John G. Thompson , title=A conjugacy theorem for E₈ , doi=10.1016/0021-8693(76)90235-0 , mr=0399193 , year=1976 , journal=Journal of Algebra , issn=0021-8693 , volume=38 , issue=2 , pages=525–530, doi-access=free

External links

MathWorld: Thompson group

Atlas of Finite Group Representations: Thompson group

Sporadic groups