Thompson group (finite)
Encyclopedia
In the mathematical field of group theory
, the Thompson group Th, found by and constructed by , is a sporadic
simple group
of order
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(F3). 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).
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(F3).)
The Schur multiplier
and the outer automorphism group
of the Thompson group are both trivial.
The Thompson group contains the Dempwolff group
as a maximal subgroup.
found the 16 classes of maximal subgroups of the Thompson group, as follows:
Group theory
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...
, the Thompson group Th, found by and constructed by , is a sporadic
Sporadic group
In the mathematical field of group theory, a sporadic group is one of the 26 exceptional groups in the classification of finite simple groups. A simple group is a group G that does not have any normal subgroups except for the subgroup consisting only of the identity element, and G itself...
simple group
Simple group
In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and the quotient group, and the process can be repeated...
of order
Order (group theory)
In group theory, a branch of mathematics, the term order is used in two closely related senses:* The order of a group is its cardinality, i.e., the number of its elements....
- 215 · 310 · 53 · 72 · 13 · 19 · 31
- = 90745943887872000
- ≈ 9 · 1016
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(F3). The subgroup preserving the Lie bracket (over the integers) is a maximal subgroup of the Thompson group called the Dempwolff group
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 25·GL5 of GL5 by its natural module of order 25...
(which unlike the Thompson group is a subgroup of the compact Lie group E8).
The centralizer of an element of order 3 of type 3C in the Monster group
Monster group
In the mathematical field of group theory, the Monster group M or F1 is a group of finite order:...
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
Vertex operator algebra
In mathematics, a vertex operator algebra is an algebraic structure that plays an important role in conformal field theory and related areas of physics...
over the field with 3 elements. (This vertex operator algebra contains the E8 Lie algebra over F3, giving the embedding of Th into E8(F3).)
The Schur multiplier
Schur multiplier
In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2 of a group G.It was introduced by in his work on projective representations.-Examples and properties:...
and the outer automorphism group
Outer automorphism group
In mathematics, the outer automorphism group of a group Gis the quotient Aut / Inn, where Aut is the automorphism group of G and Inn is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted Out...
of the Thompson group are both trivial.
The Thompson group contains the Dempwolff group
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 25·GL5 of GL5 by its natural module of order 25...
as a maximal subgroup.
found the 16 classes of maximal subgroups of the Thompson group, as follows:
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