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

In 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.

## History

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).

## 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 $T_{3C}(\tau)$ (),

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

and j(τ) is the j-function. Let $r_n$ = 1, 248, 4123, 27000, 27000, 30628, 30875, 61256, 85995, 85995, 147250,...() be the degrees of irreducible representations of Th. Then,

\begin{align} 1 &= r_1\\ 248&= r_2\\ 4124&= r_1+r_3\\ 34752&= r_1+r_3+r_6\\ 213126&= r_1+2r_2+r_3+r_8+r_{11}\\ \end{align}

and so on.

## Maximal subgroups

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:

• $2_+^{1+8}\cdot A_9$,
• $2^5\cdot L_5(2)$,
• $(3\times G_2(3)): 2$,
• $(3^3\times 3_+^{1+2})\cdot 3_+^{1+2}: 2S_4$,
• $3^2\cdot 3^7:2S_4$,
• $(3\times 3^4:2\cdot A_6): 2$,
• $5_+^{1+2}: 4S_4$,
• $5^2:GL_2(5)$,
• $7^2:(3\times 2S_4)$,
• $31:15$,
• ${}^3D_4(2): 3$,
• $U_3(8): 6$,
• $L_2(19)$,
• $L_3(3)$,
• $M_{10}$,
• $S_5$.