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

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.

In group theory, a branch of mathematics, the term order is used in two unrelated senses:

## Contents

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).

John Griggs Thompson is a mathematician at the University of Florida noted for his work in the field of finite groups. He was awarded the Fields Medal in 1970, the Wolf Prize in 1992 and the 2008 Abel Prize.

Geoffrey Charles Smith, MBE is a British mathematician. He is Senior Lecturer in Mathematics at the University of Bath and current professor in residence at Wells Cathedral School.

In mathematical finite group theory, the Dempwolff group is a finite group of order 319979520 = 215·32·5·7·31, that is the unique nonsplit extension of by its natural module of order . The uniqueness of such a nonsplit extension was shown by Dempwolff (1972), and the existence by Thompson (1976), who showed using some computer calculations of Smith (1976) that the Dempwolff group is contained in the compact Lie group as the subgroup fixing a certain lattice in the Lie algebra of , and is also contained in the Thompson sporadic group (the full automorphism group of this lattice) as a maximal subgroup.

## 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).

In the area of modern algebra known as group theory, the Monster group M (also known as the Fischer–Griess Monster, or the Friendly Giant) is the largest sporadic simple group, having order

In mathematics, a vertex operator algebra (VOA) is an algebraic structure that plays an important role in two-dimensional conformal field theory and string theory. In addition to physical applications, vertex operator algebras have proven useful in purely mathematical contexts such as monstrous moonshine and the geometric Langlands correspondence.

The Schur multiplier and the outer automorphism group of the Thompson group are both trivial.

In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group of a group G. It was introduced by Issai Schur (1904) in his work on projective representations.

In mathematics, the outer automorphism group of a group, G, is the quotient, Aut(G) / Inn(G), where Aut(G) is the automorphism group of G and Inn(G) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted Out(G). If Out(G) is trivial and G has a trivial center, then G is said to be complete.

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

In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group M and modular functions, in particular, the j function. The term was coined by John Conway and Simon P. Norton in 1979.

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

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

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