# 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*

## 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 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 |