In the area of modern algebra known as

group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...

, the Thompson group ''Th'' is a

sporadic simple group In the mathematical classification of finite simple groups, there are a number of groups which do not fit into any infinite family. These are called the sporadic simple groups, or the sporadic finite groups, or just the sporadic groups. A simpl ...

order Order, ORDER or Orders may refer to: * A socio-political or established or existing order, e.g. World order, Ancien Regime, Pax Britannica * Categorization, the process in which ideas and objects are recognized, differentiated, and understood ...

: 90,745,943,887,872,000 : = 2¹⁵3¹⁰5³7²131931 : ≈ 9.

History

''Th'' is one of the 26 sporadic groups and was found by and constructed by . They constructed it as the

automorphism group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...

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 In mathematics, specifically in group theory, the phrase ''group of Lie type'' usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with values in a finite field. The phra ...

E₈(3). The subgroup preserving the Lie bracket (over the integers) is a maximal subgroup of the Thompson group called the

Dempwolff group 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 2^\mathrm_(\mathbb_) of \mathrm_(\mathbb_) by its natural module of order 2^5. The uniquen ...

(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 In the area of abstract 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; it has order : : = 2463205976112133171923293 ...

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 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 usef ...

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 full normalizer of a 3C element in the Monster group is S₃ × Th, so Th centralizes 3 involutions alongside the 3-cycle. These involutions are centralized by the Baby monster group, which therefore contains Th as a subgroup. The

Schur multiplier In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2(G, \Z) of a group ''G''. It was introduced by in his work on projective representations. Examples and properties The Schur multiplier \ope ...

and the

outer automorphism group In mathematics, the outer automorphism group of a group, , is the quotient, , where is the automorphism group of and ) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted . If is trivial and has ...

of the Thompson group are both trivial.

Generalized monstrous moonshine

Conway and Norton suggested in their 1979 paper that

monstrous moonshine 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 initial numerical observation was made by John McKay in 1978, ...

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 In mathematics, Felix Klein's -invariant or function is a modular function of weight zero for the special linear group \operatorname(2,\Z) defined on the upper half-plane of complex numbers. It is the unique such function that is holomorphic a ...

Maximal subgroups

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

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 ''Journal of Algebra'' (ISSN 0021-8693) is an international mathematical research journal in algebra. An imprint of Academic Press, it is published by Elsevier Elsevier ( ) is a Dutch academic publishing company specializing in scientific, te ...

, 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