# Binary von Dyck group

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

The binary von Dyck group or binary triangle group with parameters $(p,q,r)$ is defined as follows:

$\langle a,b,c \mid a^p = b^q = c^r = abc \rangle$.

This group is sometimes denoted $\Gamma(p,q,r)$.

In the particular case where $q = r = 2$ or $(p,q,r) = (2,3,5),(2,3,4),(2,3,3)$, the center of the group, which is generated by $abc$, has order two, and the quotient is the von Dyck group. For other values of $(p,q,r)$, the center may not have finite order.

Further information: Center of binary von Dyck group has order two

## Relation with other groups

The quotient of the binary von Dyck group by the subgroup generated by $abc$ is the von Dyck group with parameters $(p,q,r)$. In many cases of interest, the element $abc$ has order two and the subgroup generated by it is precisely the center of the binary von Dyck group.

Each of the von Dyck groups that arise as finite subgroups of the special orthogonal group $SO(3,\R)$ have corresponding binary von Dyck groups of interest: