# Von Dyck group

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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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

## 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  or 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