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 mathematics, a sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups.
A simple group is a group ''G'' that does not have any normal subgroups except for the trivial group and ''G'' itself. The ...
of
order
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...
: 2
153
105
37
2131931
: = 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 E
8. 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 phras ...
E
8(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 E
8).
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
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 E
8 Lie algebra over F
3, giving the embedding of ''Th'' into E
8(3).
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 \oper ...
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 a t ...
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
(),
:
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''
* 2
5 · ''L''
5(2) This is 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 ...
* (3 x ''G''
2(3)) : 2
* (3
3 × 3
+1+2) · 3
+1+2 : 2''S''
4
* 3
2 · 3
7 : 2''S''
4
* (3 × 3
4 : 2 · ''A''
6) : 2
* 5
+1+2 : 4''S''
4
* 5
2 : ''GL''
2(5)
* 7
2 : (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 E
8 , 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 groupAtlas of Finite Group Representations: Thompson group
Sporadic groups