Von Dyck group
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This article defines a family of groups whose members are parametrized by tuples of natural numbers. In other words, for every tuple of natural numbers, there is a unique corresponding group (upto isomorphism) in that family
Contents
Definition
Definition in terms of presentation
For natural numbers , the von Dyck group
is defined by the following presentation:
where denotes the identity element.
This is a subgroup of index two in the triangle group, but some people use the term triangle group for the von Dyck group.
Geometric description
Given natural numbers , consider a triangle with sides
on a suitable simply connected Riemannian surface (i.e., a suitable model for Euclidean or non-Euclidean geometry). The von Dyck group is the group generated by rotations about the vertices of the triangle by angles of
,
,
respectively.
The three types
Spherical von Dyck groups
The triple in this case satisfies:
,
The solutions to which are , and
.
This is the spherical case, with the model being the unit sphere in three-dimensional space, and the corresponding von Dyck groups are termed spherical von Dyck groups. Spherical von Dyck groups are subgroups of the special orthogonal group , because
is precisely the group of orientation-preserving isometries of the sphere. All of these turn out to be finite subgroups of
, and these also turn out to be the only finite subgroups of
, a fact that follows from Euler's theorem and some additional work. The finiteness can also be viewed as a consequence of the fact that the sphere is compact and simply connected. Further information: Classification of finite subgroups of SO(3,R)
Euclidean von Dyck groups
The triple in this case satisfies:
,
for which the only solutions are and
, i.e., the right isosceles triangle and the equilateral triangle in the usual Euclidean plane.
Both of these give wallpaper groups, and neither is finite.
Hyperbolic von Dyck groups
,
for which there are infinitely many solutions. The model for this is the hyperbolic plane.
Particular cases
Smallest parameter | Middle parameter | Largest parameter | Common name for group | Group order | Symmetry object |
---|---|---|---|---|---|
1 | ![]() |
![]() |
Cyclic group | ![]() |
Regular polygon, symmetries in ![]() |
2 | 2 | ![]() |
Dihedral group | ![]() |
Regular polygon, symmetries in ![]() ![]() |
2 | 3 | 3 | Alternating group:A4 | ![]() |
Regular tetrahedron, symmetries in ![]() |
2 | 3 | 4 | Symmetric group:S4 | ![]() |
Cube or octahedron, symmetries in ![]() |
2 | 3 | 5 | Alternating group:A5 | ![]() |
Icosahedron or dodecahedron, symmetries in ![]() |