Coxeter group
This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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Definition
Symbol-free definition
A group is termed a Coxeter group if it can be equipped with a finite presentation given by the following relations:
- The square of every generator
- For some of the pairwise products of the generators, a certain power of that pairwise product
A presentation of this kind is termed a Coxeter presentation. Often, the term Coxeter group is used for the group along with a specific choice of Coxeter presentation.
Definition with symbols
A group is termed a Coxeter group if it can be equipped with a finite presentation with generators
and relations:
-
-
where
is a symmetric function of
and
(for distinct
and
)
Alternatively we can consider a symmetric matrix with the diagonal entries being
and simply require that for each
and
(not necessarily distinct)
. Note that we allow the entries
to be
.
Such a presentation is termed a Coxeter presentation and the matrix of s is termed a Coxeter matrix. Often, the term Coxeter group is used for a Coxeter group along with a specific choice of Coxeter presentation.
Particular cases
Number of generators | Form of Coxeter matrix | Common name for the group | Comment |
---|---|---|---|
2 | ![]() |
dihedral group of degree ![]() ![]() |
|
2 | ![]() |
Klein four-group | |
2 | ![]() |
symmetric group of degree three | |
2 | ![]() |
dihedral group of order eight | |
2 | ![]() |
dihedral group of order ten | |
2 | ![]() |
dihedral group of order twelve | |
2 | ![]() |
dihedral group of order sixteen | |
3 | ![]() |
triangle group with parameters ![]() | |
3 | ![]() |
elementary abelian group of order eight | |
3 | ![]() |
Direct product of dihedral group of degree ![]() ![]() | |
3 | ![]() |
symmetric group of degree four | |
3 | ![]() |
direct product of S4 and Z2 | |
3 | ![]() |
direct product of A5 and Z2 | |
3 | ![]() |
(7,3,2)-triangle group | this group is infinite. |
![]() |
1s on diagonal, 3s on superdiagonal and subdiagonal, 2s elsewhere. | symmetric group of degree ![]() |
Metaproperties
Direct products
This group property is direct product-closed, viz., the direct product of an arbitrary (possibly infinite) family of groups each having the property, also has the property
View other direct product-closed group properties
A direct product of Coxeter groups is a Coxeter group. The Coxeter matrix for the direct product is obtained by taking the block concatenation of the Coxeter matrices for the individual groups and then replacing the off-diagonal zero blocks by blocks with all entries equal to 2.